LIBRARY 

UNIVERSITY  OP 
CALIFORNIA 

SANWKJO 


A   SURVEY   OF 
SYMBOLIC   LOGIC 


BY 

C.   I.   LEWIS 


UNIVERSITY  OF  CALIFORNIA  PRESS 

BERKELEY 

1918 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER,  PA. 


TABLE    OF    CONTENTS 

PREFACE v 

CHAPTER  I.    THE  DEVELOPMENT  OF  SYMBOLIC  LOGIC.  1 
SECTION        I.    The  Scope  of  Symbolic  Logic.    Symbolic  Logic 
and    Logistic.     Summary    Account    of    their 

Development 1 

SECTION      II.     Leibniz 5 

SECTION     III.    From  Leibniz  to  De  Morgan  and  Boole 18 

SECTION     IV.    De  Morgan 37 

SECTION       V.    Boole 51 

SECTION     VI.    Jevons 72 

SECTION   VII.    Peirce 79 

SECTION  VIII.    Developments  since  Peirce 107 

CHAPTER  II.    THE   CLASSIC,   OR  BOOLE-SCHRODER  AL- 
GEBRA OF  LOGIC 118 

SECTION     I.     General  Character  of  the  Algebra.    The  Postulates 

and  their  Interpretation 118 

SECTION    II.     Elementary  Theorems 122 

SECTION  III.    General  Properties  of  Functions 132 

SECTION  IV.     Fundamental  Laws  of  the  Theory  of  Equations .  .' .  144 

SECTION    V.     Fundamental  Laws  of  the  Theory  of  Inequations.  166 
SECTION  VI.     Note  on  the  Inverse  Operations,   "Subtraction" 

and  "Division" 173 

CHAPTER  III.    APPLICATIONS  OF  THE  BOOLE-SCHRODER 

ALGEBRA 175 

SECTION     I.    Diagrams  for  the  Logical  Relations  of  Classes ....  175 

SECTION    II.    The  Application  to  Classes 184 

SECTION  III.     The  Application  to  Propositions 213 

SECTION  IV.    The  Application  to  Relations 219 

CHAPTER  IV.    SYSTEMS    BASED    ON    MATERIAL    IMPLI- 
CATION   222 

SECTION     I.    The  Two- Valued  Algebra 222 

Hi 


IV 


Table  of  Contents 


SECTION  II.  The  Calculus  of  Prepositional  Functions.  Func- 
tions of  One  Variable 232 

SECTION  III.     Prepositional  Functions  of  Two  or  More  Variables .   246 
SECTION  IV.    Derivation  of  the  Logic  of  Classes  from  the  Calcu- 
lus of  Prepositional  Functions 260 

SECTION    V.    The  Logic  of  Relations 269 

SECTION  VI.    The  Logic  of  Principia  Mathematica 279 

CHAPTER  V.    THE  SYSTEM  OF  STRICT  IMPLICATION...  291 
SECTION     I.     Primitive  Ideas,  Primitive  Propositions,  and  Im- 
mediate Consequences 292 

SECTION    II.    Strict  Relations  and  Material  Relations 299 

SECTION  III.    The  Transformation  {-/-} 306 

SECTION  IV.  Extensions  of  Strict  Implication.  The  Calculus 
of  Consistencies  and  the  Calculus  of  Ordinary 

Inference 316 

SECTION    V.    The  Meaning  of  "Implies" 324 

CHAPTER  VI.    SYMBOLIC  LOGIC,  LOGISTIC,  AND  MATHE- 
MATICAL METHOD 340 

SECTION  I.  General  Character  of  the  Logistic  Method.  The 

"Orthodox"  View 340 

SECTION  II.  Two  Varieties  of  Logistic  Method :  Peano's  Formu- 
laire  and  Principia  Mathematica.  The  Nature 
of  Logistic  Proof 343 

SECTION  III.  A  "Heterodox"  View  of  the  Nature  of  Mathe- 
matics and  of  Logistic 354 

SECTION  IV.    The  Logistic  Method  of  Kempe  and  Royce 362 

SECTION    V.     Summary  and  Conclusion 367 

APPENDIX.    TWO  FRAGMENTS  FROM  LEIBNIZ 373 

BIBLIOGRAPHY 389 

INDEX.  .  407 


PREFACE 

The  student  who  has  completed  some  elementary  study  of  symbolic 
logic  and  wishes  to  pursue  the  subject  further  finds  himself  in  a  discouraging 
situation.  He  has,  perhaps,  mastered  the  contents  of  Venn's  Symbolic 
Logic  or  Couturat's  admirable  little  book,  The  Algebra  of  Logic,  or  the 
chapters  concerning  this  subject  in  Whitehead's  Universal  Algebra.  If  he 
read  German  with  sufficient  ease,  he  may  have  made  some  excursions  into 
Schroder's  Vorlesungen  iiber  die  Algebra  der  Logik.  These  all  concern  the 
classic,  or  Boole-Schroder  algebra,  and  his  knowledge  of  symbolic  logic  is 
probably  confined  to  that  system.  His  further  interest  leads  him  almost 
inevitably  to  Peano's  Formulaire  de  Mathematiques,  Principia  Mathematica 
of  Whitehead  and  Russell,  and  the  increasingly  numerous  shorter  studies 
of  the  same  sort.  And  with  only  elementary  knowledge  of  a  single  kind  of 
development  of  a  small  branch  of  the  subject,  he  must  attack  these  most 
difficult  and  technical  of  treatises,  in  a  new  notation,  developed  by  methods 
which  are  entirely  novel  to  him,  and  bristling  with  logico-metaphysical 
difficulties.  If  he  is  bewildered  and  searches  for  some  means  of  further 
preparation,  he  finds  nothing  to  bridge  the  gap.  Schroder's  work  would 
be  of  most  assistance  here,  but  this  was  written  some  twenty-five  years 
ago;  the  most  valuable  studies  are  of  later  date,  and  radically  new  methods 
have  been  introduced. 

What  such  a  student  most  needs  is  a  comprehensive  survey  of  the  sub- 
ject— one  which  will  familiarize  him  with  more  than  the  single  system 
which  he  knows,  and  will  indicate  not  only  the  content  of  other  branches 
and  the  alternative  methods  of  procedure,  but  also  the  relation  of  these  to 
the  Boole-Schroder  algebra  and  to  one  another.  The  present  book  is  an 
attempt  to  meet  this  need,  by  bringing  within  the  compass  of  a  single 
volume,  and  reducing  to  a  common  notation  (so  far  as  possible),  the  most 
important  developments  of  symbolic  logic.  If,  in  addition  to  this,  some 
of  the  requirements  of  a  "handbook"  are  here  fulfilled,  so  much  the  better. 

But  this  survey  does  not  pretend  to  be  encyclopedic.  A  gossipy  recital 
of  results  achieved,  or  a  superficial  account  of  methods,  is  of  no  more  use 
in  symbolic  logic  than  in  any  other  mathematical  discipline.  What  is 
presented  must  be  treated  in  sufficient  detail  to  afford  the  possibility  of  real 
insight  and  grasp.  This  aim  has  required  careful  selection  of  material. 


vi  Preface 

The  historical  summary  in  Chapter  I  attempts  to  follow  the  main  thread 
of  development,  and  no  reference,  or  only  passing  mention,  is  given  to 
those  studies  which  seem  not  to  have  affected  materially  the  methods  of 
later  researches.  In  the  remainder  of  the  book,  the  selection  has  been 
governed  by  the  same  purpose.  Those  topics  comprehension  of  which 
seems  most  essential,  have  been  treated  at  some  length,  while  matters  less 
fundamental  have  been  set  forth  in  outline  only,  or  omitted  altogether. 
My  own  contribution  to  symbolic  logic,  presented  in  Chapter  V,  has  not 
earned  the  right  to  inclusion  here;  in  this,  I  plead  guilty  to  partiality. 
The  discussion  of  controversial  topics  has  been  avoided  whenever  possible 
and,  for  the  rest,  limited  to  the  simpler  issues  involved.  Consequently, 
the  reader  must  not  suppose  that  any  sufficient  consideration  of  these 
questions  is  here  given,  though  such  statements  as  are  made  will  be,  I  hope, 
accurate.  Particularly  in  the  last  chapter,  on  "Symbolic  Logic,  Logistic, 
and  Mathematical  Method  ",  it  is  not  possible  to  give  anything  like  an 
adequate  account  of  the  facts.  That  would  require  a  volume  at  least  the 
size  of  this  one.  Rather,  I  have  tried  to  set  forth  the  most  important  and 
critical  considerations — somewhat  arbitrarily  and  dogmatically,  since  there 
is  not  space  for  argument — and  to  provide  such  a  map  of  this  difficult  terri- 
tory as  will  aid  the  student  in  his  further  explorations. 

Proofs  and  solutions  in  Chapters  II,  III,  and  IV  have  been  given  very 
fully.  Proof  is  of  the  essence  of  logistic,  and  it  is  my  observation  that  stu- 
dents— even  those  with  a  fair  knowledge  of  mathematics — seldom  command 
the  technique  of  rigorous  demonstration.  In  any  case,  this  explicitness  can 
do  no  harm,  since  no  one  need  read  a  proof  which  he  already  understands. 

I  am  indebted  to  many  friends  and  colleagues  for  valuable  assistance  in 
preparing  this  book  for  publication :  to  Professor  W.  A.  Merrill  for  emenda- 
tions of  my  translation  of  Leibniz,  to  Professor  J.  H.  McDonald  and 
Dr.  B.  A.  Bernstein  for  important  suggestions  and  the  correction  of  certain 
errors  in  Chapter  II,  to  Mr.  J.  C.  Rowell,  University  Librarian,  for  assistance 
in  securing  a  number  of  rare  volumes,  and  to  the  officers  of  the  University 
Press  for  their  patient  helpfulness  in  meeting  the  technical  difficulties  of 
printing  such  a  book.  Mr.  Shirley  Quimby  has  read  the  whole  book  in 
manuscript,  eliminated  many  mistakes,  and  verified  most  of  the  proofs. 

But  most  of  all,  I  am  indebted  to  my  friend  and  teacher,  Josiah  Royce, 
who  first  aroused  my  interest  in  this  subject,  and  who  never  failed  to  give 
me  encouragement  and  wise  counsel.  Much  that  is  best  in  this  book  is 
due  to  him.  C.  I.  LEWIS. 

BERKELEY,  July  10,  1917. 


CHAPTER    I 

THE  DEVELOPMENT  OF  SYMBOLIC  LOGIC 

I.    THE  SCOPE  OF  SYMBOLIC  LOGIC.    SYMBOLIC  LOGIC  AND  LOGISTIC. 
SUMMARY  ACCOUNT  OF  THEIR  DEVELOPMENT 

The  subject  with  which  we  are  concerned  has  been  variously  referred 
to  as  "symbolic  logic",  "logistic",  "algebra  of  logic",  "calculus  of  logic", 
"mathematical  logic",  "algorithmic  logic",  and  probably  by  other  names. 
And  none  of  these  is  satisfactory.  We  have  chosen  "symbolic  logic" 
because  it  is  the  most  commonly  used  in  England  and  in  this  country,  and 
because  its  signification  is  pretty  well  understood.  Its  inaccuracy  is 
obvious:  logic  of  whatever  sort  uses  symbols.  We  are  concerned  only 
with  that  logic  which  uses  symbols  in  certain  specific  ways — those  ways 
which  are  exhibited  generally  in  mathematical  procedures.  In  particular, 
logic  to  be  called  "symbolic"  must  make  use  of  symbols  for  the  logical 
relations,  and  must  so  connect  various  relations  that  they  admit  of  "trans- 
formations" and  "operations",  according  to  principles  which  are  capable 
of  exact  statement. 

If  we  must  give  some  definition,  we  shall  hazard  the  following:  Symbolic 
Logic  is  the  development  of  the  most  general  principles  of  rational  pro- 
cedure, in  ideographic  symbols,  and  in  a  form  which  exhibits  the  connection 
of  these  principles  one  with  another.  Principles  which  belong  exclusively 
to  some  one  type  of  rational  procedure — e.  g.  to  dealing  with  number  and 
quantity — are  hereby  excluded,  and  generality  is  designated  as  one  of  the 
marks  of  symbolic  logic. 

Such  general  principles  are  likewise  the  subject  matter  of  logic  in  any 
form.  To  be  sure,  traditional  logic  has  never  taken  possession  of  more 
than  a  small  portion  of  the  field  which  belongs  to  it.  The  modes  of  Aristotle 
are  unnecessarily  restricted.  As  we  shall  have  occasion  to  point  out,  the 
reasons  for  the  syllogistic  form  are  psychological,  not  logical :  the  syllogism, 
made  up  of  the  smallest  number  of  propositions  (three),  each  with  the  small- 
est number  of  terms  (two),  by  which  any  generality  of  reasoning  can  be 
attained,  represents  the  limitations  of  human  attention,  not  logical  necessity. 
To  regard  the  syllogism  as  indispensable,  or  as  reasoning  par  excellence,  is 
2  1 


2  A  Survey  of  Symbolic  Logic 

the  apotheosis  of  stupidity.  And  the  procedures  of  symbolic  logic,  not 
being  thus  arbitrarily  restricted,  may  seem  to  mark  a  difference  of  subject 
matter  between  it  and  the  traditional  logic.  But  any  such  difference  is 
accidental,  not  essential,  and  the  really  distinguishing  mark  of  symbolic 
logic  is  the  approximation  to  a  certain  form,  regarded  as  ideal.  There  are 
all  degrees  of  such  approximation ;  hence  the  difficulty  of  drawing  any  hard 
and  fast  line  between  symbolic  and  other  logic. 

But  more  important  than  the  making  of  any  such  sharp  distinction  is 
the  comprehension  of  that  ideal  of  form  upon  which  it  is  supposed  to 
depend.  The  most  convenient  method  which  the  human  mind  has  so  far 
devised  for  exhibiting  principles  of  exact  procedure  is  the  one  which  we 
call,  in  general  terms,  mathematical.  The  important  characteristics  of 
this  form  are:  (1)  the  use  of  ideograms  instead  of  the  phonograms  of 
ordinary  language;  (2)  the  deductive  method — which  may  here  be  taken 
to  mean  simply  that  the  greater  portion  of  the  subject  matter  is  derived 
from  a  relatively  few  principles  by  operations  which  are  "exact":  and 
(3)  the  use  of  variables  having  a  definite  range  of  significance. 

Ideograms  have  two  important  advantages  over  phonograms.  In  the 
first  place,  they  are  more  compact,  +  than  "plus",  3  than  "three",  etc. 
This  is  no  inconsiderable  gain,  since  it  makes  possible  the  presentation  of  a 
formula  in  small  enough  compass  so  that  the  eye  may  apprehend  it  at  a 
glance  and  the  image  of  it  (in  visual  or  other  terms)  may  be  retained  for 
reference  with  a  minimum  of  effort.  None  but  a  very  thoughtless  person, 
or  one  without  experience  of  the  sciences,  can  fail  to  understand  the  enor- 
mous advantage  of  such  brevity.  In  the  second  place,  an  ideographic 
notation  is  superior  to  any  other  in  precision.  Many  ideas  which  are 
quite  simply  expressible  in  mathematical  symbols  can  only  with  the  greatest 
difficulty  be  rendered  in  ordinary  language.  Without  ideograms,  even 
arithmetic  would  be  difficult,  and  higher  branches  impossible. 

The  deductive  method,  by  which  a  considerable  array  of  facts  is  sum- 
marized in  a  few  principles  from  which  they  can  be  derived,  is  much  more 
than  the  mere  application  of  deductive  logic  to  the  subject  matter  in 
question.  It  both  requires  and  facilitates  such  an  analysis  of  the  whole 
body  of  facts  as  will  most  precisely  exhibit  their  relations  to  one  another. 
In  fact,  any  other  value  of  the  deductive  form  is  largely  or  wholly  fictitious. 

The  presentation  of  the  subject  matter  of  logic  in  this  mathematical 
form  constitutes  what  we  mean  by  symbolic  logic.  Hence  the  essential 
characteristics  of  our  subject  are  the  following:  (1)  Its  subject  matter  is 


The  Development  of  Symbolic  Logic  3 

the  subject  matter  of  logic  in  any  form — that  is,  the  principles  of  rational 
or  reflective  procedure  in  general,  as  contrasted  with  principles  which 
belong  exclusively  to  some  particular  branch  of  such  procedure.  (2)  Its 
medium  is  an  ideographic  symbolism,  in  which  each  separate  character 
represents  a  relatively  simple  and  entirely  explicit  concept.  And,  ideally, 
all  non-ideographic  symbolism  or  language  is  excluded.  (3)  Amongst  the 
ideograms,  some  will  represent  variables  (the  "terms"  of  the  system) 
having  a  definite  range  of  significance.  Although  it  is  non-essential,  in 
any  system  so  far  developed  the  variables  will  represent  "individuals", 
or  classes,  or  relations,  or  propositions,  or  " propositional  functions",  or 
they  will  represent  ambiguously  some  two  or  more  of  these.  (4)  Any 
system  of  symbolic  logic  will  be  developed  deductively — that  is,  the  whole 
body  of  its  theorems  will  be  derived  from  a  relatively  few  principles,  stated 
in  symbols,  by  operations  which  are,  or  at  least  can  be,  precisely  formulated. 

We  have  been  at  some  pains  to  make  as  clear  as  possible  the  nature  of 
symbolic  logic,  because  its  distinction  from  "ordinary"  logic,  on  the  one 
hand,  and,  on  the  other,  from  any  mathematical  discipline  in  a  sufficiently 
abstract  form,  is  none  too  definite.  It  will  be  further  valuable  to  comment 
briefly  on  some  of  the  alternative  designations  for  the  subject  which  have 
been  mentioned. 

"Logistic"  would  not  have  served  our  purpose,  because  "logistic"  is 
commonly  used  to  denote  symbolic  logic  together  with  the  application  of 
its  methods  to  other  symbolic  procedures.  Logistic  may  be  defined  as 
the  science  which  deals  with  types  of  order  as  such.  It  is  not  so  much  a 
subject  as  a  method.  Although  most  logistic  is  either  founded  upon  or 
makes  large  use  of  the  principles  of  symbolic  logic,  still  a  science  of  order 
in  general  does  not  necessarily  presuppose,  or  begin  with,  symbolic  logic. 
Since  the  relations  of  symbolic  logic,  logistic,  and  mathematics  are  to  be 
the  topic  of  the  last  chapter,  we  may  postpone  any  further  discussion  of 
that  matter  here.  We  have  mentioned  it  only  to  make  clear  the  meaning 
which  "logistic"  is  to  have  in  the  pages  which  follow.  It  comprehends 
symbolic  logic  and  the  application  of  such  methods  as  symbolic  logic  exempli- 
fies to  other  exact  procedures.  Its  subject  matter  is  not  confined  to  logic. 

"Algebra  of  logic"  is  hardly  appropriate  as  the  general  name  for  our 
subject,  because  there  are  several  quite  distinct  algebras  of  logic,  and 
because  symbolic  logic  includes  systems  which  are  not  true  algebras  at  all. 
"The  algebra  of  logic"  usually  means  that  system  the  foundations  of 
which  were  laid  by  Leibniz,  and  after  him  independently  by  Boole,  and 


4  A  Survey  of  Symbolic  Logic 

which  was  completed  by  Schroder.  We  shall  refer  to  this  system  as  the 
"Boole-Schroder  Algebra  ". 

"Calculus"  is  a  more  general  term  than  "algebra".  By  a  "calculus" 
will  be  meant,  not  the  whole  subject,  but  any  single  system  of  assumptions 
and  their  consequences. 

The  program  both  for  symbolic  logic  and  for  logistic,  in  anything  like  a 
clear  form,  was  first  sketched  by  Leibniz,  though  the  ideal  of  logistic  seems 
to  have  been  present  as  far  back  as  Plato's  Republic.1  Leibniz  left  frag- 
mentary developments  of  symbolic  logic,  and  some  attempts  at  logistic 
which  are  prophetic  but  otherwise  without  value.  After  Leibniz,  the  two 
interests  somewhat  diverge.  Contributions  to  symbolic  logic  were  made  by 
Ploucquet,  Lambert,  Castillon  and  others  on  the  continent.  This  type  of 
research  interested  Sir  William  Hamilton  and,  though  his  own  contribution 
was  slight  and  not  essentially  novel,  his  papers  were,  to  some  extent  at 
least,  responsible  for  the  renewal  of  investigations  in  this  field  which  took 
place  in  England  about  1845  and  produced  the  work  of  De  Morgan  and 
Boole.  Boole  seems  to  have  been  ignorant  of  the  work  of  his  continental 
predecessors,  which  is  probably  fortunate,  since  his  own  beginning  has 
proved  so  much  more  fruitful.  Boole  is,  in  fact,  the  second  founder  of  the 
subject,  and  all  later  work  goes  back  to  his.  The  main  line  of  this  develop- 
ment runs  through  Jevons,  C.  S.  Peirce,  and  MacColl  to  Schroder  whose 
Vorlesungen  uber  die  Algebra  der  Logik  (Vol.  I,  1890)  marks  the  perfection 
of  Boole's  algebra  and  the  logical  completion  of  that  mode  of  procedure. 

In  the  meantime,  interest  in  logistic  persisted  on  the  continent  and 
was  fostered  by  the  growing  tendency  to  abstractness  and  rigor  in  mathe- 
matics and  by  the  hope  for  more  general  methods.  Hamilton's  quaternions 
and  the  Ausdehnungslehre  of  Grassmann,  which  was  recognized  as  a  con- 
tinuation of  the  work  begun  by  Leibniz,  contributed  to  this  end,  as  did  also 
the  precise  logical  analyses  of  the  nature  of  number  by  Cantor  and  Dedekind. 
Also,  the  elimination  from  "modern  geometry"  of  all  methods  of  proof 
dependent  upon  "intuitions  of  space"  or  "construction"  brought  that 
subject  within  the  scope  of  logistic  treatment,  and  in  1889  Peano  provided 
such  a  treatment  in  I  Principii  di  Geometria.  Frege's  works,  from  the 
Begri/sschrift  of  1879  to  the  Grundgesetze  der  Arithmetik  (Vol.  I,  1893; 
Vol.  II,  1903)  provide  a  comprehensive  development  of  arithmetic  by  the 
logistic  method. 

1  See  the  criticisms  of  contemporary  mathematics  and  the  program  for  the  dialectic 
or  philosophic  development  of  mathematics  in  Bk.  vi,  Step.  510-11  and  Philebus,  Step.  56-57. 


The  Development  of  Symbolic  Logic  5 

In  1894,  Peano  and  his  collaborators  began  the  publication  of  the 
Formulaire  de  Mathematiques,  in  which  all  branches  of  mathematics  were  to 
be  presented  in  the  universal  language  of  logistic.  In  this  work,  symbolic 
logic  and  logistic  are  once  more  brought  together,  since  the  logic  presented 
in  the  early  sections  provides,  in  a  way,  the  method  by  which  the  other 
branches  of  mathematics  are  developed.  The  Formulaire  is  a  monumental 
production.  But  its  mathematical  interests  are  as  much  encyclopedic  as 
logistic,  and  not  all  the  possibilities  of  the  method  are  utilized  or  made 
clear.  It  remained  for  Whitehead  and  Russell,  in  Principia  Mathematica, 
to  exhibit  the  perfect  union  of  symbolic  logic  and  the  logistic  method  in 
mathematics.  The  publication  of  this  work  undoubtedly  marks  an  epoch 
in  the  history  of  the  subject.  The  tendencies  marked  in  the  development 
of  the  algebra  of  logic  from  Boole  to  Schroder,  in  the  development  of  the 
algebra  of  relatives  from  De  Morgan  to  Schroder,  and  in  the  foundations 
for  number  theory  of  Cantor  and  Dedekind  and  Frege,  are  all  brought 
together  here.2  Further  researches  will  most  likely  be  based  upon  the 
formulations  of  Principia  Mathematica. 

We  must  now  turn  back  and  trace  in  more  detail  the  development  of 
symbolic  logic.3  A  history  of  the  subject  will  not  be  attempted,  if  by 
history  is  meant  the  report  of  facts  for  their  own  sake.  Rather,  we  are 
interested  in  the  cumulative  process  by  which  those  results  which  most 
interest  us  today  have  come  to  be.  Many  researches  of  intrinsic  value, 
but  lying  outside  the  main  line  of  that  development,  will  of  necessity  be 
neglected.  Reference  to  these,  so  far  as  we  are  acquainted  with  them,  will 
be  found  in  the  bibliography.4 

II.    LEIBNIZ 

The  history  of  symbolic  logic  and  logistic  properly  begins  with  Leibniz.5 
In  the  New  Essays  on  the  Human  Understanding,  Philalethes  is  made  to 
say : 6  "I  begin  to  form  for  myself  a  wholly  different  idea  of  logic  from 
that  which  I  formerly  had.  I  regarded  it  as  a  scholar's  diversion,  but  I 
now  see  that,  in  the  way  you  understand  it,  it  is  like  a  universal  mathe- 

2  Perhaps  we  should  add  "and  the  modern  development  of  abstract  geometry,  as  by 
Hilbert,  Fieri,  and  others",  but  the  volume  of  Principia  which  is  to  treat  of  geometry  has 
not  yet  appeared. 

3  The  remainder  of  this  chapter  is  not  essential  to  an  understanding  of  the  rest  of  the 
book.     But  after  Chapter  i,  historical  notes  and  references  are  generally  omitted. 

4  Pp.  389-406. 

6  Leibniz  regards  Raymond  Lully,  Athanasius  Kircher,  John  Wilkins,  and  George 
Dalgarno  (see  Bibliography)  as  his  predecessors  in  this  field.  But  their  writings  contain 
little  which  is  directly  to  the  point. 

8  Bk.  iv,  Chap,  xvn,  §  9. 


6  A  Survey  of  Symbolic  Logic 

matics."  As  this  passage  suggests,  Leibniz  correctly  foresaw  the  general 
character  which  logistic  was  to  have  and  the  problems  it  would  set  itself 
to  solve.  But  though  he  caught  the  large  outlines  of  the  subject  and 
actually  delimited  the  field  of  work,  he  failed  of  any  clear  understanding 
of  the  difficulties  to  be  met,  and  he  contributed  comparatively  little  to 
the  successful  working  out  of  details.  Perhaps  this  is  characteristic  of  the 
man.  But  another  explanation,  or  partial  explanation,  is  possible.  Leibniz 
expected  that  the  whole  of  science  would  shortly  be  reformed  by  the  appli- 
cation of  this  method.  This  was  a  task  clearly  beyond  the  powers  of  any 
one  man,  who  could,  at  most,  offer  only  the  initial  stimulus  and  general 
plan.  And  so,  throughout  his  life,  he  besought  the  assistance  of  learned 
societies  and  titled  patrons,  to  the  end  that  this  epoch-making  reform  might 
be  instituted,  and  never  addressed  himself  very  seriously  to  the  more 
limited  tasks  which  he  might  have  accomplished  unaided.7  Hence  his 
studies  in  this  field  are  scattered  through  the  manuscripts,  many  of  them 
still  unedited,  and  out  of  five  hundred  or  more  pages,  the  systematic  results 
attained  might  be  presented  in  one-tenth  the  space.8 

Leibniz's  conception  of  the  task  to  be  accomplished  altered  somewhat 
during  his  life,  but  two  features  characterize  all  the  projects  which  he 
entertained:  (1)  a  universal  medium  ("universal  language"  or  "rational 
language"  or  "universal  characteristic")  for  the  expression  of  science; 
and  (2)  a  calculus  of  reasoning  (or  "universal  calculus")  designed  to  display 
the  most  universal  relations  of  scientific  concepts  and  to  afford  some  sys- 
tematic abridgment  of  the  labor  of  rational  investigation  in  all  fields,  much 
as  mathematical  formulae  abridge  the  labor  of  dealing  with  quantity  and 
number.  "The  true  method  should  furnish  us  with  an  Ariadne's  thread, 
that  is  to  say,  with  a  certain  sensible  and  palpable  medium,  which  will 
guide  the  mind  as  do  the  lines  drawn  in  geometry  and  the  formulae  for 
operations  which  are  laid  down  for  the  learner  in  arithmetic."9 

This  universal  medium  is  to  be  an  ideographic  language,  each  single 
character  of  which  will  represent  a  simple  concept.  It  will  differ  from 
existing  ideographic  languages,  such  as  Chinese,  through  using  a  combina- 

7  The  editor's  introduction  to  "Scientia  Generalis.     Characteristica"  in  Gerhardt's 
Philosophischen  Schriften  von  Leibniz  (Berlin,  1890),  vn,  gives  an  excellent  account  of 
Leibniz's  correspondence  upon  this  topic,  together  with  other  material  of  historic  interest. 
(Work  hereafter  cited  as  G.  Phil.) 

8  See  Gerhardt,  op.  dt.  especially  iv  and  vn.      But  Couturat,  La  logique  de  Leibniz 
(1901),  gives  a  survey  which  will  prove  more  profitable  to  the  general  reader  than  any 
study  of  the  sources. 

9  Letter  to  Galois,  1677,  G.  Phil,  vn,  21. 


The  Development  of  Symbolic  Logic  7 

tion  of  symbols,  or  some  similar  device,  for  a  compound  idea,  instead  of 
having  a  multiplicity  of  characters  corresponding  to  the  variety  of  things. 
So  that  while  Chinese  can  hardly  be  learned  in  a  lifetime,  the  universal 
characteristic  may  be  mastered  in  a  few  weeks.10  The  fundamental  char- 
acters of  the  universal  language  will  be  few  in  number,  and  will  represent 
the  "alphabet  of  human  thought":  "The  fruit  of  many  analyses  will  be  the 
catalogue  of  ideas  which  are  simple  or  not  far  from  simple."  n  With  this 
catalogue  of  primitive  ideas — this  alphabet  of  human  thought — the  whole 
of  science  is  to  be  reconstructed  in  such  wise  that  its  real  logical  organiza- 
tion will  be  reflected  in  its  symbolism. 

In  spite  of  fantastic  expression  and  some  hyperbole,  we  recognize  here 
the  program  of  logistic.  If  the  reconstruction  of  all  science  is  a  project  too 
ambitious,  still  we  should  maintain  the  ideal  possibility  and  the  desirability 
of  such  a  reconstruction  of  exact  science  in  general.  And  the  ideographic 
language  finds  its  realization  in  Peano's  Formulaire,  in  Principia  Mathe- 
matica,  and  in  all  successful  applications  of  the  logistic  method. 

Leibniz  stresses  the  importance  of  such  a  language  for  the  more  rapid 
and  orderly  progress  of  science  and  of  human  thought  in  general.  The 
least  effect  of  it  "...  will  be  the  universality  and  communication  of 
different  nations.  Its  true  use  will  be  to  paint  not  the  word  .  .  .  but  the 
thought,  and  to  speak  to  the  understanding  rather  than  to  the  eyes.  .  .  . 
Lacking  such  guides,  the  mind  can  make  no  long  journey  without  losing 
its  way  .  .  .  :  with  such  a  medium,  we  could  reason  in  metaphysics 
and  in  ethics  very  much  as  we  do  in  geometry  and  in  analytics,  because  the 
characters  would  fix  our  ideas,  which  are  otherwise  too  vague  and  fleeting 
in  such  matters  in  which  the  imagination  cannot  help  us  unless  it  be  by 
the  aid  of  characters."12  The  lack  of  such  a  universal  medium  prevents 
cooperation.  "The  human  race,  considered  in  its  relation  to  the  sciences 
which  serve  our  welfare,  seems  to  me  comparable  to  a  troop  which  marches 
in  confusion  in  the  darkness,  without  a  leader,  without  order,  without  any 
word  or  other  signs  for  the  regulation  of  their  march  and  the  recognition  of 
one  another.  Instead  of  joining  hands  to  guide  ourselves  and  make  sure 
of  the  road,  we  run  hither  and  yon  and  interfere  with  one  another."13 

The  "alphabet  of  human  thought"  is  more  visionary.  The  possibility 
of  constructing  the  whole  of  a  complex  science  from  a  few  primitive  con- 

10  Letter  to  the  Duke  of  Hanover,  1679  (?),  G.  Phil,  vii,  24-25. 

11  G.  Phil,  vii,  84. 

12  G.  Phil.,  vii,  21. 
18  G.  Phil,  vii,  157. 


8  A  Survey  of  Symbolic  Logic 

cepts  is,  indeed,  real — vide  the  few  primitives  of  Principia  Mathematica. 
But  we  should  today  recognize  a  certain  arbitrariness  in  the  selection  of 
these,  though  an  arbitrariness  limited  by  the  nature  of  the  subject.  The 
secret  of  Leibniz's  faith  that  these  primitive  concepts  are  fixed  in  the  nature 
of  things  will  be  found  in  his  conception  .of  knowledge  and  of  proof.  He 
believes  that  all  predicates  are  contained  in  the  (intension  of  the)  subject 
and  may  be  discovered  by  analysis.  Similarly,  all  truths  which  are  not 
absolutely  primitive  and  self-evident  admit  of  reduction  by  analysis  into 
such  absolutely  first  truths.  And  finally,  only  one  real  definition  of  a 
thing — "real"  as  opposed  to  "nominal" — is  possible;14  that  is,  the  result 
of  the  correct  analysis  of  any  concept  is  unambiguously  predetermined  in 
the  concept  itself. 

The  construction,  from  such  primitives,  of  the  complex  concepts  of 
the  various  sciences,  Leibniz  speaks  of  as  "synthesis"  or  "invention", 
and  he  is  concerned  about  the  "art  of  invention".  But  while  the  result  of 
analysis  is  always  determined,  and  only  one  analysis  is  finally  correct, 
synthesis,  like  inverse  processes  generally,  has  no  such  predetermined 
character.  In  spite  of  the  frequent  mention  of  the  subject,  the  only  im- 
portant suggestions  for  this  art  have  to  do  with  the  provision  of  a  suitable 
medium  and  of  a  calculus  of  reasoning.  To  be  sure  there  are  such  obvious 
counsels  as  to  proceed  from  the  simple  to  the  complex,  and  in  the  early 
essay,  De  Arte  Combinatoria,  there  are  studies  of  the  possible  permutations 
and  combinations  or  "syntheses"  of  fundamental  concepts,  but  the  author 
later  regarded  this  study  as  of  little  value.  And  in  Initia  et  Specimina 
Scientice  nova  Generalis,  he  says  that  the  utmost  which  we  can  hope  to 
accomplish  at  present,  toward  the  general  art  of  invention,  is  a  perfectly 
orderly  and  finished  reconstruction  of  existing  science  in  terms  of  the 
absolute  primitives  which  analysis  reveals.15  After  two  hundred  years, 
we  are  still  without  any  general  method  by  which  logistic  may  be  used  in 
fields  as  yet  unexplored,  and  we  have  no  confidence  in  any  absolute  primi- 
tives for  such  investigation. 

The  calculus  of  reasoning,  or  universal  calculus,  is  to  be  the  instrument 
for  the  development  and  manipulation  of  systems  in  the  universal  language, 
and  it  is  to  get  its  complete  generality  from  the  fact  that  all  science  will  be 
expressed  in  the  ideographic  symbols  of  that  universal  medium.  The 
calculus  will  consist  of  the  general  principles  of  operating  with  such  ideo- 

14  See  G.  Phil,  vn,  194,  footnote. 

15  G.  Phil,  vn,  84. 


The  Development  of  Symbolic  Logic  9 

graphic  symbols:  "All  our  reasoning  is  nothing  but  the  relating  and  sub- 
stituting of  characters,  whether  these  characters  be  words  or  marks  or 
images. " 16  Thus  while  the  characteristica  universalis  is  the  project  of  the 
logistic  treatment  of  science  in  general,  the  universal  calculus  is  the  pre- 
cursor of  symbolic  logic. 

The  plan  for  this  universal  calculus  changed  considerably  with  the 
development  of  Leibniz's  thought,  but  he  speaks  of  it  always  as  a  mathe- 
matical procedure,  and  always  as  more  general  than  existing  mathematical 
methods.17  The  earliest  form  suggested  for  it  is  one  in  which  the  simple 
concepts  are  to  be  represented  by  numbers,  and  the  operations  are  to  be 
merely  those  of  arithmetical  multiplication,  division,  and  factoring.  When, 
later,  he  abandons  this  plan  of  procedure,  he  speaks  of  a  general  calculus 
which  will  be  concerned  with  what  we  should  nowadays  describe  as  "types 
of  order" — wTith  combinations  which  are  absolute  or  relative,  symmetrical 
or  unsymmetrical,  and  so  on.1*  His  latest  studies  toward  such  a  calculus 
form  the  earliest  presentation  of  what  we  now  call  the  "algebra  of  logic". 
But  it  is  doubtful  if  Leibniz  ever  thought  of  the  universal  calculus  as 
restricted  to  our  algebra  of  logic:  we  can  only  say  that  it  was  intended  to 
be  the  science  of  mathematical  and  deductive  form  in  general  (it  is  doubtful 
whether  induction  was  included),  and  such  as  to  make  possible  the  appli- 
cation of  the  analytic  method  of  mathematics  to  all  subjects  of  which 
scientific  knowledge  is  possible. 

Of  the  various  studies  to  this  end  our  chief  interest  will  be  in  the  early 
essay,  De  Arte  Combinatorial  and  in  the  fragments  which  attempt  to 
develop  an  algebra  of  logic.20 

Leibniz  wrote  De  Arte  Combinatoria  when  he  was,  in  his  own  words, 
mx  egressus  ex  Ephebis,  and  before  he  had  any  considerable  knowledge  of 
mathematics.  It  was  published,  he  tells  us,  without  his  knowledge  or 
consent.  The  intention  of  the  work,  as  indicated  by  its  title,  is  to  serve  the 
general  art  of  rational  invention,  as  the  author  conceived  it.  As  has  been 
mentioned,  it  seems  that  this  end  is  to  be  accomplished  by  a  complete 
analysis  of  concepts  of  the  topic  under  investigation  and  a  general  survey 
of  the  possibilities  of  their  combination.  A  large  portion  of  the  essay  is 
concerned  with  the  calculation  of  the  possible  forms  of  this  and  that  type 

16  G.  Phil,  vn,  31. 

17  See  New  Essays  on  the  Human  Understanding,  Bk.  iv,  Chap,  xvn,  §§  9-13. 

18  See  G.  Phil,  vn,  31,  IQSff.,  and  204. 

19  G.  Phil.,  iv,  35-104.     Also  Gerhardt,  Leibnizens  malhematische  Schriften  (1859),  v, 
1-79. 

20  Sdentia  Generalis.     Characteristica,  xv-xx,  G.  Phil.,  vn. 


10  A  Survey  of  Symbolic  Logic 

of  logical  construct:  the  various  dyadic,  triadic,  etc.,  complexes  which 
can  be  formed  with  a  given  number  of  elements;  of  the  moods  and  figures 
of  the  syllogism;  of  the  possible  predicates  of  a  given  subject  (the  com- 
plexity of  the  subject  as  a  concept  being  itself  the  key  to  the  predicates 
which  can  be  analyzed  out  of  it);  of  the  number  of  propositions  from  a 
given  number  of  subjects,  given  number  of  predicate  relations,  and  given 
number  of  quaestiones ; 21  of  the  variations  of  order  with  a  given  number  of 
terms,  and  so  on.  In  fact  so  much  space  is  occupied  with  the  computation 
of  permutations  and  combinations  that  some  of  his  contemporaries  failed 
to  discover  any  more  important  meaning  of  the  essay,  and  it  is  most  fre- 
quently referred  to  simply  as  a  contribution  to  combinatorial  analysis.22 

Beyond  this  the  significance  of  the  essay  lies  in  the  attempt  to  devise  a 
symbolism  which  will  preserve  the  relation  of  analyzable  concepts  to  their 
primitive  constituents.  The  particular  device  selected  for  this  purpose — 
representation  of  concepts  by  numbers — is  unfortunate,  but  the  attempt 
itself  is  of  interest.  Leibniz  makes  application  of  this  method  to  geometry 
and  suggests  it  for  other  sciences.23  In  the  geometrical  illustration,  the 
concepts  are  divided  into  classes.  Class  1  consists  of  concepts  or  terms 
regarded  as  elementary  and  not  further  analyzable,  each  of  which  is  given  a 
number.  Thereafter,  the  number  is  the  symbol  of  that  concept.  Class  2 
consists  of  concepts  analyzable  into  (definable  in  terms  of)  those  of  Class  1. 
By  the  use  of  a  fractional  notation,  both  the  class  to  which  a  concept 
belongs  and  its  place  in  that  class  can  be  indicated  at  once.  The  denomi- 
nator indicates  the  number  of  the  class  and  the  numerator  is  the  number  of 
the  concept  in  that  class.  Thus  the  concept  numbered  7  in  Class  2  is 
represented  by  7/2.  Class  3  consists  of  concepts  definable  in  terms  of 
those  in  Class  1  and  Class  2,  and  so  on.  By  this  method,  the  complete 
analysis  of  any  concept  is  supposed  to  be  indicated  by  its  numerical  symbol.24 

21  Leibniz  tells  us  that  he  takes  this  problem  from  the  Ars  Magna  of  Raymond  Lully. 
See  G.  Phil.,  v,  62. 

22  See  letter  to  Tschirnhaus,  1678,  Gerhardt,  Math.,  iv,  451-63.     Cf.  Cantor,  Geschichle 
d.  Math.,  in,  39  ff. 

23  See  the  Synopsis,  G.  Phil.,  iv,  30-31. 

24  See  Couturat,  op.  cit.,  appended  Note  vi,  p.  554.$'. 

The  concepts  are  arranged  as  follows  (G.  Phil.,  iv,  70-72): 

"Classis  I;  1.  Punctum,  2.  Spatium,  3.  intervallum,  4.  adsitum  seu  contiguum,  5.  dis- 
situm  seu  distans,  6.  Terminus  seu  quae  distant,  7.  Insitum,  8.  inclusum  (v.g.  centrum  est 
insitum  circulo,  inclusum  peripheriae),  9.  Pars,  10.  Totum,  11.  idem,  12.  diversum,  13.  unum, 
14.  Numerus,  etc.  etc.  [There  are  twenty-seven  numbered  concepts  in  this  class.] 

"Classis  II;  1.  Quantitas  est  14  T&V  9  (15).  [Numbers  enclosed  in  parentheses  have 
their  usual  arithmetical  significance,  except  that  (15)  signifies  'an  indefinite  number'.] 
2.  Includens  est  6.10.  III.  1.  Intervallum  est  2.3.10.  2.  Aequale  A  rf/s  11.  J.  3.  Continuum 
est  A  ad  B,  si  TOV  A  r)  9  est  4  et  7  TU  B.;  etc.  etc." 


The  Development  of  Symbolic  Logic  1 1 

In  point  of  fact,  the  analysis  (apart  from  any  merely  geometrical  defects) 
falls  far  short  of  being  complete.  Leibniz  uses  not  only  the  inflected  Greek 
article  to  indicate  various  relations  of  concepts  but  also  modal  inflections 
indicated  by  et,  si,  quod,  quam  faciunt,  etc. 

In  later  years  Leibniz  never  mentions  this  work  without  apologizing  for 
it,  yet  he  always  insists  that  its  main  intention  is  sound.  This  method 
of  assuming  primitive  ideas  which  are  arbitrarily  symbolized,  of  introducing 
other  concepts  by  definition  in  terms  of  these  primitives  and,  at  the  same 
time,  substituting  a  single  symbol  for  the  complex  of  defining  symbols — 
this  is,  in  fact,  the  method  of  logistic  in  general.  Modern  logistic  differs 
from  this  attempt  of  Leibniz  most  notably  in  two  respects:  (1)  modern 
logistic  would  insist  that  the  relations  whereby  two  or  more  concepts  are 
united  in  a  definition  should  be  analyzed  precisely  as  the  substantives  are 
analyzed;  (2)  while  Leibniz  regards  his  set  of  primitive  concepts  as  the 
necessary  result  of  any  proper  analysis,  modern  logistic  would  look  upon 
them  as  arbitrarily  chosen.  Leibniz's  later  work  looks  toward  the  elimina- 
tion of  this  first  difference,  but  the  second  represents  a  conviction  from 
which  he  never  departed. 

At  a  much  later  date  come  various  studies  (not  in  Gerhardt),  which 
attempt  a  more  systematic  use  of  number. and  of  mathematical  operations 
in  logic.25  Simple  and  primitive  concepts,  Leibniz  now  proposes,  should  be 
symbolized  by  prime  numbers,  and  the  combination  of  two  concepts  (the 
qualification  of  one  term  by  another)  is  to  be  represented  by  their  product. 
Thus  if  3  represent  "rational"  and  7  "animal",  "man"  will  be  21.  No 
prime  number  will  enter  more  than  once  into  a  given  combination — a 
rational  rational  animal,  or  a  rational  animal  animal,  is  simply  a  rational 
animal.  Thus  logical  synthesis  is  represented  by  arithmetical  multipli- 
cation: logical  analysis  by  resolution  into  prime  factors.  The  analysis  of 
"man",  21,  would  be  accomplished  by  finding  its  prime  factors,  "rational", 
3,  and  "animal",  7.  In  accordance  with  Leibniz's  conviction  that  all 
knowledge  is  analytic  and  all  valid  predicates  are  contained  in  the  subject, 
the  proposition  "All  S  is  P"  will  be  true  if  the  number  which  represents 
the.  concept  S  is  divisible  by  that  which  represents  P.  Accordingly  the 

25  Dated  April,  1679.  Couturat  (op.  tit.,  p.  326,  footnote)  gives  the  titles  of  these 
as  follows:  "Elemenla  Characteristicae  Universalis  (Collected  manuscripts  of  Leibniz  in 
the  Hanover  Library,  PHIL.,  v,  8  b);  Calculi  universalis  Elemenla  (PHIL.,  v,  8  c);  Calculi 
universalis  investigaliones  (PHIL.,  v,  8  d);  Modus  examinandi  consequentias  per  numeros 
(PHIL.,  v,  8  e);  Regulae  ex  quibus  de  bonilate  consequentiarum  formisque  et  modis  syllogis- 
morum  categoricum  judicari  potest  per  numeros  (PHIL.,  v,  8f)."  These  fragments,  with 
many  others,  are  contained  in  Couturat's  Opuscules  et  fragments  inedits  de  Leibniz. 


12  A  Survey  of  Symbolic  Logic 

universal  affirmative  proposition  may  be  symbolized  by  S/P  =  y  or  S  =  Py 
(where  y  is  a  whole  number).  By  the  plan  of  this  notation,  Py  will  represent 
some  species  whose  "difference",  within  the  genus  P,  is  y.  Similarly  Sx 
will  represent  a  species  of  S.  Hence  the  particular  affirmative,  "Some 
S  is  P,"  may  be  symbolized  by  Sx  =  Py,  or  S/P  =  y/x.  Thus  the  uni- 
versal is  a  special  case  of  the  particular,  and  the  particular  will  always  be 
true  when  the  universal  is  true. 

There  are  several  objections  to  this  scheme.  In  the  first  place,  it 
presumes  that  any  part  of  a  class  is  a  species  within  the  class  as  genus. 
This  is  far-fetched,  but  perhaps  theoretically  defensible  on  the  ground 
that  any  part  which  can  be  specified  by  the  use  of  language  may  be  treated 
as  a  logical  species.  A  worse  defect  lies  in  the  fact  that  Sx  =  Py  will 
always  be  true.  For  a  given  S  and  P,  we  can  always  find  x  and  y  which 
will  satisfy  the  equation  Sx  =  Py.  If  no  other  choice  avails,  let  x  =  P, 
or  some  multiple  of  P,  and  y  =  S,  or  some  multiple  of  S.  " Angel-man" 
=  "man-angel"  although  no  men  are  angels.  "Spineless  man"  =  "ra- 
tional invertebrate",  but  it  is  false  that  some  men  are  invertebrates.  A 
third  difficulty  arises  because  of  the  existential  import  of  the  particular — 
a  difficulty  which  later  drew  Leibniz's  attention.  If  the  particular  affirma- 
tive is  true,  then  for  some  x  and  y-,  Sx  =  Py.  The  universal  negative  should, 
then,  be  Sx  =f  Py.  And  since  the  universal  affirmative  is  S  =  Py,  the 
particular  negative  should  be  S  =t=  Py-  But  this  symbolism  would  be 
practically  unworkable  because  the  inequations  would  have  to  be  verified 
for  all  values  of  x  and  y.  Also,  as  we  have  noted,  the  equality  Sx  =  Py 
will  always  hold  and  Sx  ={=  Py,  where  x  and  y  are  arbitrary,  will  never  be 
true. 

Such  difficulties  led  Leibniz  to  complicate  his  symbolism  still  further, 
introducing  negative  numbers  and  finally  using  a  pair  of  numbers,  one 
positive  and  one  negative,  for  each  concept.  But  this  scheme  also  breaks 
down,  and  the  attempt  to  represent  concepts  by  numbers  is  thereafter 
abandoned. 

Of  more  importance  to  symbolic  logic  are  the  later  fragments  included 
in  the  plans  for  an  encyclopedia  which  should  collect  and  arrange  all  known 
science  as  the  proper  foundation  for  future  work.26  Leibniz  cherished  the 

26  G.  Phil.,  vn,  xvi-xx.  Of  these,  xvi,  without  title,  states  rules  for  inference  in 
terms  of  inclusion  and  exclusion;  Difficultates  quaedam  logicae  treats  of  subalternation 
and  conversion  and  of  the  symbolic  expression  for  various  types  of  propositions;  xvin, 
Specimen  Calculi  universalis  with  its  addenda  and  marginal  notes,  gives  the  general  prin- 
ciples of  procedure  for  the  universal  calculus;  xix,  with  the  title  Non  inelegans  specimen 


The  Development  of  Symbolic  Logic  13 

notion  that  this  should  be  developed  in  terms  of  the  universal  characteristic. 
In  these  fragments,  the  relations  of  equivalence,  inclusion,  and  qualification 
of  one  concept  by  another,  or  combination,  are  defined  and  used.  These 
relations  are  always  considered  in  intension  when  it  is  a  question  of  apply- 
ing the  calculus  to  formal  logic.  "Equivalence"  is  the  equivalence  of 
concepts,  not  simply  of  two  classes  which  have  the  same  members;  "for  A 
to  include  B  or  B  to  be  included  in  A  is  to  affirm  the  predicate  B  universally 
of  the  subject  A".27  However,  Leibniz  evidently  considers  the  calculus 
to  have  many  applications,  and  he  thinks  out  the  relations  and  illustrates 
them  frequently  in  terms  of  extensional  diagrams,  in  which  A,  B,  etc.,  are 
represented  by  segments  of  a  right  line.  Although  he  preferred  to  treat 
logical  relations  in  intension,  he  frequently  states  that  relations  of  intension 
are  easily  transformed  into  relations  of  extension.  If  A  is  included  in  B 
in  intension,  B  is  included  in  A  in  extension;  and  a  calculus  may  be  inter- 
preted indifferently  as  representing  relations  of  concepts  in  intension  or 
relations  of  individuals  and  classes  in  extension.  Also,  the  inclusion  rela- 
tion may  be  interpreted  as  the  relation  of  an  antecedent  proposition  to  a 
consequent  proposition.  The  hypothesis  A  includes  its  consequence  B, 
just  as  the  subject  A  includes  the  predicate  B.28  This  accords  with  his 
frequently  expressed  conviction  that  all  demonstration  is  analysis.  Thus 
these  studies  are  by  no  means  to  be  confined  to  the  logic  of  intension.  As 
one  title  suggests,  they  are  studies  demonstrandi  in  abstractis. 

demonstrandi  in  abstractis  struck  out,  and  xx,  without  title,  are  deductive  developments 
of  theorems  of  symbolic  logic,  entirely  comparable  with  later  treatises. 

The  place  of  symbolic  logic  in  Leibniz's  plans  for  the  Encyclopedia  is  sufficiently 
indicated  by  the  various  outlines  which  he  has  left.  In  one  of  these  (G.  Phil.,  vu,  49), 
divisions  1-6  are  of  an  introductory  nature,  after  which  come: 

"7.  De  scientiarum  instauratione,  ubi  de  Systematibus  et  Repertoriis,  et  de  Encyclo- 
paedia demonstrativa  codenda. 

"8.  Elementa  veritatis  aeternae,  et  de  arte  demonstrandi  in  omnibus  disciplinis  ut  in 
Mathesi. 

"9.  De  novo  quodam  Calculo  generali,  cujus  ope  tollantur  omnes  disputationes  inter 
eos  qui  in  ipsum  consenserit;  est  Cabala  sapient um. 

"10.  De  Arte  Inveniendi. 

"11.  De  Synthesi  seu  Arte  combinatoria. 

"12.  DeAnalysi. 

"  13.  De  Combinatoria  speciali,  seu  scientia  formarum,  sive  qualitatum  in  genere  (de 
Characterismis)  sive  de  simili  et  dissimili. 

"14.  De  Analysi  speciali  seu  scientia  quantitatum  in  genere  seu  de  magno  et  parvo. 

"  15.  De  Mathesi  generali  ex  duabus  praecedentibus  composita." 

Then  various  branches  of  mathematics,  astronomy,  physics,  biological  science,  medi- 
cine, psychology,  political  science,  economics,  military  science,  jurisprudence,  and  natural 
theology,  in  the  order  named. 

27  G.  Phil,  vii,  208. 

28  "Generales  Inquisitiones"   (1686):  see  Couturat,  Opuscuks  etc.,  pp.  356-99. 


14  A  Survey  of  Symbolic  Logic 

It  is  a  frequent  remark  upon  Leibniz's  contributions  to  logic  that  he 
failed  to  accomplish  this  or  that,  or  erred  in  some  respect,  because  he 
chose  the  point  of  view  of  intension  instead  of  that  of  extension.  The 
facts  are  these:  Leibniz  too  hastily  presumed  a  complete,  or  very  close, 
analogy  between  the  various  logical  relations.  It  is  a  part  of  his  sig- 
nificance for  us  that  he  sought  such  high  generalizations  and  believed  in 
their  validity.  He  preferred  the  point  of  view  of  intension,  or  connotation, 
partly  from  habit  and  partly  from  rationalistic  inclination.  As  a  conse- 
quence, wherever  there  is  a  discrepancy  between  the  intensional  and  ex- 
tensional  points  of  view,  he  is  likely  to  overlook  it,  and  to  follow  the  former. 
This  led  him  into  some  difficulties  which  he  might  have  avoided  by  an 
opposite  inclination  and  choice  of  example,  but  it  also  led  him  to  make 
some  distinctions  the  importance  of  which  has  since  been  overlooked  and 
to  avoid  certain  difficulties  into  which  his  commentators  have  fallen.29 

In  Difficulties  quaedam  logicae,  Leibniz  shows  that  at  last  he  recognizes 
the  difficulty  in  connecting  the  universal  and  the  corresponding  particular. 
He  sees  also  that  this  difficulty  is  connected  with  the  disparity  between  the 
intensional  point  of  view  and  the  existential  import  of  particular  proposi- 
tions. In  the  course  of  this  essay  he  formulates  the  symbolism  for  the  four 
propositions  in  two  different  ways.  The  first  formulation  is: 30 

Univ.  aff.;  All  A  is  B:  AB  =  A,  or  A  non-5  does  not  exist. 
Part,  neg.;  Some  A  is  not  B;  AB  4=  A,  or  A  non-5  exists. 
Univ.  neg.;  No  A  is  B;  AB  does  not  exist. 
Part,  aff.;  Some  A  is  B;  AB  exists. 

AB  =  A  and  AB  4=  A  may  be  interpreted  as  relations  of  intension  or  of 
extension  indifferently.  If  all  men  are  mortal,  the  intension  of  "mortal 
man"  is  the  same  as  the  intension  of  "man",  and  likewise  the  class  of 
mortal  men  is  identical  in  extent  with  the  class  of  men.  The  statements 
concerning  existence  are  obviously  to  be  understood  in  extension  only. 
The  interpretation  here  put  upon  the  propositions  is  identically  that  of 
contemporary  symbolic  logic.  With  these  expressions,  Leibniz  infers  the 
subaltern  and  the  converse  of  the  subaltern,  from  a  given  universal,  by 

29  For  example,  it  led  him  to  distinguish  the  merely  non-existent  from  the  absurd,  or 
impossible,  and  the  necessarily  true  from  the  contingent.     See  G.  Phil.,  vn,  231,  foot- 
note; and  "Specimen  certitudinis  seu  de  conditionibus,"  Dutens,  Leibnitii  Opera,  iv, 
Part  in,  pp.  92  ff.,  also  Couturat,  La  Logique  de  Leibniz,  p.  348,  footnote,  and  p.  353, 
footnote. 

30  G.  Phil.,  viz,  212. 


The  Development  of  Symbolic  Logic  15 

means  of  the  hypothesis  that  the  subject,  A,  exists.  Later  in  the  essay,  he 
gives  another  set  of  expressions  for  the  four  propositions : 31 

All  ,4  is  B:  AB  =  A. 

Some  A  is  not  B:  AB  =(=  A. 

No  A  is  B:  AB  does  not  exist,  or  AB  4=  AB  Ens. 

Some  A  is  B :  AB  exists,  or  AB  =  AB  Ens. 

In  the  last  two  of  these,  AB  before  the  sign  of  equality  represents  the 
possible  AB's  or  the  AB  "in  the  region  of  ideas";  "AB  Ens"  represents 
existing  AB's,  or  actual  members  of  the  class  AB.  (Read  AB  Ens,  "  AB 
which  exists".)  AB  =  AB  Ens  thus  represents  the  fact  that  the  class  AB 
has  members;  AB  4=  AB  Ens,  that  the  class  AB  has  no  members.  A 
logical  species  of  the  genus  A,  "some  A",  may  be  represented  by  YA; 
YA  Ens  will  represent  existing  members  of  that  species,  or  "some  exist- 
ing A".  Leibniz  correctly  reasons  that  if  AB  =  A  (All  A  is  J5),  YAB 
=  YA  (Some  A  is  B);  but  if  AB  4=  A,  it  does  not  follow  that  YAB  4=  YA, 
for  if  Y  =  B,  YAB  =  YA.  Again,  if  AB  4=  AB  Ens  (No  A  is  B),  YAB 
4=  YAB  Ens  (It  is  false  that  some  A  is  5);  but  if  AB  =  AB  Ens  (Some 
A  is  B),  YAB  =  YAB  Ens  does  not  follow,  because  Y  could  assume  values 
incompatible  with  A  and  B.  For  example,  some  men  are  wise,  but  it  does 
not  follow  that  foolish  men  are  foolish  wise  persons,  because  "foolish"  is 
incompatible  with  "wise".32  The  distinction  here  between  AB,  a  logical 
division  of  A  or  of  B,  and  AB  Ens,  existing  AB's,  is  ingenious.  This  is 
our  author's  most  successful  treatment  of  the  relations  of  extension  and 
intension,  and  of  the  particular  to  the  universal. 

In  Specimen  calculi  universalis,  the  "principles  of  the  calculus"  are 
announced  as  follows : 33 

1)  "Whatever  is  concluded  in  terms  of  certain  variable  letters  may  be 
concluded  in  terms  of  any  other  letters  which  satisfy  the  same  conditions; 
for  example,  since  it  is  true  that  [all]  ab  is  a,  it  will  also  be  true  that  [all] 
be  is  b  and  that  [all]  bed  is  be.  .  .  . 

2)  "Transposing  letters  in  terms  changes  nothing;    for  example  ab 
coincides  with  ba,  'animal  rational'  with  'rational  animal'. 

3)  "Repetition  of  a  letter  in  the  same  term  is  useless.  .  .  . 

4)  "One  proposition  can  be  made  from  any  number  by  joining  all  the 
subjects  in  one  subject  and  all  the  predicates  in  one  predicate :  Thus,  a  is  6 
and  c  is  d  and  e  is  /,  become  ace  is  bdf.  .  .  . 

31  G.  Phil.,  vii,  213-14. 

32  G.  Phil.,  vn,  215:  the  illustration  is  mine. 

33  G.  Phil,  vn,  224-25. 


16  A  Survey  of  Symbolic  Logic 

5)  "From  any  proposition  whose  predicate  is  composed  of  more  than 
one  term,  more  than  one  proposition  can  be  made;  each  derived  proposition 
having  the  subject  the  same  as  the  given  proposition  but  in  place  of  the 
given  predicate  some  part  of  the  given  predicate.  If  [all]  a  is  bed,  then  [all] 
a  is  b  and  [all]  a  is  c  and  [all]  a  is  d. "  24 

If  we  add  to  the  number  of  these,  two  principles  which  are  announced 
under  the  head  of  "self-evident  propositions" — (1)  a  is  included  in  a; 
and  (2)  ab  is  included  in  a — we  have  here  the  most  important  of  the  funda- 
mental principles  of  symbolic  logic.  Principle  1  is  usually  qualified  by 
some  doctrine  of  the  "universe  of  discourse"  or  of  "range  of  significance", 
but  some  form  of  it  is  indispensable  to  algorithms  in  general.  The  law 
numbered  2  above  is  what  we  now  call  the  "principle  of  permutation"; 
3,  the  "principle  of  tautology";  4,  the  "principle  of  composition";  5,  the 
"principle  of  division".  And  the  two  "self-evident  propositions"  are  often 
included  in  sets  of  postulates  for  the  algebra  of  logic. 

There  remain  for  consideration  the  two  fragments  which  are  given  in 
translation  in  our  Appendix,  XIX  and  XX  of  Scientia  Generalis:  Char- 
acteristica.  The  first  of  these,  with  the  title  Non  ineleqans  specimen  demon- 
strandi  in  abstractis,  stricken  out  in  the  manuscript,  is  rather  the  more  inter- 
esting. Here  the  relation  previously  symbolized  by  AB  or  ab  is  represented 
by  A+B.  And  A+B  =  L  signifies  that  A  is  contained  or  included  in 
(est  in)  B.  A  scholium  attached  to  the  definition  of  this  inclusion  relation 
distinguishes  it  from  the  part-whole  relation.  Comparison  of  this  and 
other  passages  shows  that  Leibniz  uses  the  inclusion  relation  to  cover 
(1)  the  relation  of  a  member  of  the  class  to  the  class  itself;  (2)  the  relation 
of  a  species,  or  subclass,  to  its  genus — a  relation  in  extension;  (3)  the  rela- 
tion of  a  genus  to  one  of  its  species — a  relation  of  intension.  The  first  of 
these  is  our  e-relation;  (2)  is  the  inclusion  relation  of  the  algebra  of  logic; 
and  (3)  is  the  analogous  relation  of  intension.  Throughout  both  these 
fragments,  it  is  clear  that  Leibniz  thinks  out  his  theorems  in  terms  of 
extensional  diagrams,  in  which  classes  or  concepts  are  represented  by 
segments  of  a  line,  and  only  incidently  in  terms  of  the  intension  of  concepts. 

The  different  interpretations  of  the  symbols  must  be  carefully  dis- 
tinguished. If  A  is  "rational"  and  B  is  "animal",  and  A  and  B  are  taken 
in  intension,  then  A+B  will  represent  "rational  animal".  But  if  A  and  B 
are  classes  taken  in  extension,  then  A  +  B  is  the  class  made  up  of  those 
things  which  are  either  A  or  B  (or  both).  Thus  the  inclusion  relation, 

34  4.  and  5.  are  stated  without  qualification  because  this  study  is  confined  to  the  proper- 
ties of  universal  affirmative  propositions.  4.  is  true  also  for  universal  negatives. 


The  Development  of  Symbolic  Logic  17 

A  +  B  =  L,  may  be  interpreted  either  in  intension  or  in  extension  as  "A  is 
in  L  ".  This  is  a  little  confusing  to  us,  because  we  should  nowadays  invert 
the  inclusion  relation  when  we  pass  from  intension  to  extension;  instead 
of  this,  Leibniz  changes  the  meaning  of  A  +  B  from  "both  A  and  B"  (in 
intension)  to  "either  A  or  B"  (in  extension).  If  A  is  "rational",  B  "ani- 
mal ",  and  L  "man",  then  A  +  B  =  L  is  true  in  intension,  "rational  animal" 
=  "man"  or  "rational"  is  contained  in  "man".  If  A,  B,  and  L  are  classes 
of  points,  or  segments  of  a  line,  then  A  +  B  =  L  will  mean  that  L  is  the 
class  of  points  comprising  the  points  in  A  and  the  points  in  B  (any  points 
common  to  A  and  B  counted  only  once),  or  the  segment  made  up  of 
segments  A  and  B. 

The  relation  A+B  does  not  require  that  A  and  B  should  be  mutually 
exclusive.  If  L  is  a  line,  A  and  B  may  be  overlapping  segments;  and,  in 
intension,  A  and  B  may  be  overlapping  concepts,  such  as  "triangle"  and 
"equilateral",  each  of  which  contains  the  component  "figure". 

Leibniz  also  introduces  the  relation  L  —  A,  which  he  calls  detractio. 
L  —  A  —  N  signifies  that  L  contains  A  and  that  if  A  be  taken  from  L  the 
remainder  is  N.  The  relations  [+]  and  [  — ]  are  not  true  inverses:  if 
A+B  =  L,  it  does  not  follow  that  L  —  A  =  B,  because  A  and  B  may  be 
overlapping  (in  Leibniz's  terms,  communicantia) .  If  L  —  A  =  N,  A  and  N 
must  be  mutually  exclusive  (incommunicantia) .  Hence  if  A+B  =  L  and 
A  and  B  have  a  common  part,  M,  L  —  A  =  B  —  M.  (If  the  reader  will 
take  a  line,  L,  in  which  A  and  B  are  overlapping  segments,  this  will  be 
clear.)  This  makes  the  relation  of  detractio  somewhat  confusing.  In 
extension,  L  —  A  may  be  interpreted  "  L  which  is  not  A  ".  In  intension, 
it  is  more  difficult.  Leibniz  offers  the  example:  "man"  -  "rational" 
=  "brute",  and  calls  our  attention  to  the  fact  that  "man"  •  "rational" 
is  not  "non-rational  man"  or  "man"+  "non-rational".35  In  intension,  the 
relation  seems  to  indicate  an  abstraction,  not  a  negative  qualification. 
But  there  are  difficulties,  due  to  the  overlapping  of  concepts.  Say  that 
" man"  +  "woodworking"  =  "carpenter"  and  "  man"  +  "white-skinned" 

36  G.  Phil.,  vii,  231,  footnote.  Couturat  in  commenting  on  this  (op.  tit.,  pp.  377-78) 
says: 

"Ailleurs  Leibniz  essaie  de  pr6ciser  cette  opposition  en  disant: 

'A  —A  est  Nihilum.  Sed  A  non-A  est  Absurdum.'  "Mais  il  oublie  que  le  n6ant 
(non-Ens)  n'est  pas  autre  chose  que  ce  qu'il  appelle  1'absurde  ou  1'impossible,  c'est-a-dire 
le  contradictoire. " 

It  may  be  that  Couturat,  not  Leibniz,  is  confused  on  this  point.  Non-existence  may 
be  contingent,  as  opposed  to  the  necessary  non-existence  of  the  absurd.  And  the  result  of 
abstracting  A  from  the  concept  A  seems  to  leave  merely  non-Ens,  not  absurdity. 

3 


18  A  Survey  of  Symbolic  Logic 

=  "Caucasian".  Then  " Caucasian "+" carpenter "  =  "man"  +  "white- 
skinned  "  +  "  woodworking  ".  Hence  ("  Caucasian  "  +  "  carpenter ")  —  "  car- 
penter" =  "white-skinned",  because  the  common  constituent  "man"  has 
been  abstracted  in  abstracting  "carpenter".  That  is,  the  abstraction  of 
"carpenter"  from  "Caucasian  carpenter"  leaves,  not  "Caucasian"  but 
only  that  part  of  the  concept  "Caucasian"  which  is  wholly  absent  in 
"carpenter".  We  cannot  here  say  "white-skinned  man"  because  "man" 
is  abstracted,  nor  "white-skinned  animal"  because  "animal"  is  contained 
in  "man":  we  can  only  say  "white-skinned"  as  a  pure  abstraction.  Such 
abstraction  is  difficult  to  carry  out  and  of  very  little  use  as  an  instrument 
of  logical  analysis.  Leibniz's  illustration  is  scribbled  in  the  margin  of  the 
manuscript,  and  it  seems  clear  that  at  this  point  he  was  not  thinking  out 
his  theorems  in  terms  of  intensions. 

Fragment  XX  differs  from  XIX  in  that  it  lacks  the  relation  symbolized 
by  [  — ].  This  is  a  gain  rather  than  a  loss,  both  because  of  the  difficulty  of 
interpretation  and  because  [+  ]  and  [— ]  are  not  true  inverses.  Also  XX 
is  more  carefully  developed :  more  of  the  simple  theorems  are  proved,  and 
more  illustrations  are  given.  Otherwise  the  definitions,  relations,  and 
methods  of  proof  are  the  same.  In  both  fragments  the  fundamental 
operation  by  which  theorems  are  proved  is  the  substitution  of  equivalent 
expressions. 

If  the  successors  of  Leibniz  had  retained  the  breadth  of  view  which 
characterizes  his  studies  and  aimed  to  symbolize  relations  of  a  like  generality, 
these  fragments  might  well  have  proved  sufficient  foundation  for  a  satis- 
factory calculus  of  logic. 

III.    FROM  LEIBNIZ  TO  DE  MORGAN  AND  BOOLE 

After  Leibniz,  various  attempts  were  made  to  develop  a  calculus  of 
logic.  Segner,  Jacques  Bernoulli,  Ploucquet,  Tonnies,  Lambert,  Holland, 
Castillon,  and  others,  all  made  studies  toward  this  end.  Of  these,  the 
most  important  are  those  of  Ploucquet,  Lambert  and  Castillon,  while  one 
of  Holland's  is  of  particular  interest  because  it  intends  to  be  a  calculus 
in  extension.  But  this  attempt  was  not  quite  a  success,  and  the  net  result 
of  the  others  is  to  illustrate  the  fact  that  a  consistent  calculus  of  logical 
relations  in  intension  is  either  most  difficult  or  quite  impossible. 

Of  Segner's  work  and  Ploucquet's  we  can  give  no  account,  since  no 
copies  of  these  writings  are  available.36  Venn  makes  it  clear  that  Plouc- 

36  There  seem  to  be  no  copies  of  Ploucquet's  books  in  this  country,  and  attempts  to 
secure  them  from  the  continent  have  so  far  failed. 


The  Development  of  Symbolic  Logic  19 

quet's  calculus  was  a  calculus  of  intension  and  that  it  involved  the  quanti- 
fication of  the  predicate. 

Lambert37  wrote  voluminously  on  the  subject  of  logic,  but  his  most 
important  contribution  to  symbolic  procedure  is  contained  in  the  Seeks 
Versuche  einer  Zeichenkunst  in  der  Vernunftlehre.™  These  essays  are  not 
separate  studies,  made  from  different  beginnings;  later  essays  presuppose 
those  which  precede  and  refer  to  their  theorems;  and  yet  the  development 
is  not  entirely  continuous.  Material  given  briefly  in  one  will  be  found 
set  forth  more  at  length  in  another.  And  discussion  of  more  general  prob- 
lems of  the  theory  of  knowledge  and  of  scientific  method  are  sometimes 
introduced.  But  the  important  results  can  be  presented  as  a  continuous 
development  which  follows  in  general  the  order  of  the  essays. 

Lambert  gives  the  following  list  of  his  symbols: 

The  symbol  of  equality  (Gleichgiiltigkeit)  = 

addition  (Zusetzung)  + 

abstraction  (Absonderung)  — 
opposition  (des  Gegentheils}  X 

universality  > 

particularity  < 
copula 

given  concepts  (Begri/e)  a,  b,  c,  d,  etc. 

undetermined  concepts  n,  m,  I,  etc. 

unknowns  x,  y,  z. 

the  genus  7 

the  difference  d 

The  calculus  is  developed  entirely  from  the  point  of  view  of  intension: 
the  letters  represent  concepts,  not  classes,  [  +  ]  indicates  the  union  of  two 
concepts  to  form  a  third,  [  —  ]  represents  the  withdrawal  or  abstraction  of 
some  part  of  the  connotation  of  a  concept,  while  the  product  of  a  and  b 
represents  the  common  part  of  the  two  concepts.  7  and  5  qualify  any 
term  "multiplied"  into  them.  Thus  ay  represents  the  genus  of  a,  ad  the 
difference  of  a.  Much  use  is  made  of  the  well-known  law  of  formal  logic 
that  the  concept  (of  a  given  species)  equals  the  genus  plus  the  difference. 

(1)  ay  +  ad  =  a(y  +5)  =  a 

37  Johann  Heinrich  Lambert  (1728-77),  German  physicist,  mathematician,  and  astrono- 
mer.    He  is  remembered  chiefly  for  his  development  of  the  equation  xn+px  =  q  in  an 
infinite  series,  and  his  proof,  in  1761,  of  the  irrationality  of  v. 

38  In  Logische  und    philosophische  Abhandlungen;  ed.  Joh.  Bernoulli  (Berlin,  1782), 
vol.  i. 


20  A  Survey  of  Symbolic  Logic 

ay  +  ad  is  the  definition  or  explanation  (Erklarung)  of  a.  As  immediate 
consequences  of  (1),  we  have  also 

(2)     ay  =  a  —  ad  (3)     a8  =  a  —  ay 

Lambert  takes  it  for  granted  that  [+  ]  and  [  — ]  are  strictly  inverse  opera- 
tions. We  have  already  noted  the  difficulties  of  Leibniz  on  this  point. 
If  two  concepts,  a  and  b,  have  any  part  of  their  connotation  in  common, 
then  (a  +  &)  —  b  will  not  be  a  but  only  that  part  of  a  which  does  not  belong 
also  to  b.  If  "European"  and  "carpenter"  have  the  common  part  "man", 
then  ("European"*  "carpenter")  minus  "carpenter"  is  not  "European" 
but  "European"  minus  "man".  And  [+  ]  and  [— ]  will  not  here  be  true 
inverses.  But  this  difficulty  may  be  supposed  to  disappear  where  the 
terms  of  the  sum  are  the  genus  and  difference  of  some  concept,  since  genus 
ajid  difference  may  be  supposed  to  be  mutually  exclusive.  We  shall  return 
to  this  topic  later. 

More  complex  laws  of  genus  and  difference  may  be  elicited  from  the 
fact  that  the  genus  of  any  given  a  is  also  a  concept  and  can  be  "explained," 
as  can  also  the  difference  of  a. 

(4)  a  =  a(y  +  5)2  =  ayz  +  ayd  +  ady  +  ad2 

Proof:      ay  =  ayy  +  ayd         and         ad  =  ady  +  add 
But  a  =  ay  +  ad.     Hence  Q.E.D. 

That  is  to  say:  if  one  wyish  to  define  or  explain  a,  one  need  not  stop  at 
giving  its  genus  and  difference,  but  may  define  the  genus  in  terms  of  its 
genus  and  difference,  and  define  the  difference  similarly.  Thus  a  is  equiva- 
lent to  the  genus  of  the  genus  of  a  plus  the  difference  of  the  genus  of  a  plus 
the  genus  of  the  difference  of  a  plus  the  difference  of  the  difference  of  a. 
This  may  be  called  a  "higher"  definition  or  "explanation"  of  a. 

Obviously,  this  process  of  higher  and  higher  "explanation"  may  be 
carried  to  any  length;  the  result  is  what  Lambert  calls  his  "Newtonian 
formula".  We  shall  best  understand  this  if  we  take  one  more  preliminary 
step.  Suppose  the  explanation  carried  one  degree  further  and  the  resulting 
terms  arranged  as  follows: 

a  =  a(y3  +  yyd  +  ydd  +  53) 
+  ydy +  dyd 
+  dyy  +  ddy 
The  three  possible  arrangements  of  two  y's  and  one  d  might  be  summarized 


The  Development  of  Symbolic  Logic  21 

by  3725;  the  three  arrangements  of  two  6's  and  one  7  by  3752.     With  this 
convention,  the  formula  for  an  explanation  carried  to  any  degree,  n,  is: 

n(n-l)  n(n-l}(n-2) 

(5)  a  =  a(yn  +  nyn~lb  +  -  -— — -  7*-252  +   -  -  y*~353  +  .  .  .  etc. 

—  .  O  1 

This  "Newtonian  formula"  is  a  rather  pleasant  mathematical  conceit. 
Two  further  interesting  laws  are  given: 

(6)  a  =  ad  +  ay8  +  ay28  +  ay38  +  .  .  .  etc. 
Proof :  a  =  ay  +  a8 

But  ay  =  ay2  +  ay  8 

and         a72  =  ay3  +  ay28 

ay3  =  ay4  +  ay38,  etc.  etc. 

(7)  a  =  ay  +  ady  +  a82y  +  a83y  +  .  .  .  etc. 
Proof:  a  =  ay  +  a8 

But  a8  =  a8y  +  a52 

and         ad2  =  a82y  +  ad3 

a83  =  a83y  +  a64,  etc.  etc. 

Just  as  the  genus  of  a  is  represented  by  ay,  the  genus  of  the  genus  of 
a  by  a72,  etc.,  so  a  species  of  which  a  is  genus  may  be  represented  by  ay~l, 
and  a  species  of  which  a  is  genus  of  the  genus  by  «7~2,  etc.  In  general,  as 
ayn  represents  a  genus  above  a,  so  a  species  below  a  may  be  represented  by 

a 
ay~n        or 

yn 

Similarly  ainy  concept  of  which  a  is  difference  of  the  difference  of  the  differ- 
ence .  .  .  etc.,  may  be  represented  by 

a 
aS-'         or          -~ 

Also,  just  as  a  =  a(y  +  8)n,  where  a  is  a  concept  and  a(y  +  8}n  its  "explana- 
tion", so- ^—  =  a,  where ^'ls  the  concept  and  a  the  "explanation" 

of  it. 

Certain  cautions  in  the  transformation  of  expressions,  both  with  respect 
to  "multiplication"  and  with  respect  to  "division,"  need  to  be  observed.40 

39  Seeks  Versuche,  p.  5. 

40  Ibid.,  pp.  9-10. 


22  A  Survey  of  Symbolic  Logic 

The  concept  ay2  +  ady  is  very  different  from  the  concept  (ay  +  ad~)y,  because 

(8)  (ay  +  a8)y  =  0(7+5)7  =  ay(y+d)  =  ay 

while  O72  +  a8y  is  the  genus  of  the  genus  of  a  plus  the  genus  of  the  difference 

of  a.     Also  —  7  must  be  distinguished  from  — .       -  7  is  the  genus  of  any 
7  77 

species  x  of  which  a  is  the  genus,  i.  e., 

(9)  -  7  =  a 

7 

But  07/7  is  any  species  of  which  the  genus  of  a  is  the  genus,  i.  e.,  any 
species  x  such  that  a  and  x  belong  to  the  same  genus. 

We  turn  now  to  consideration  of  the  relation  of  concepts  which  have  a 
common  part. 

Similarity  is  identity  of  properties.  Two  concepts  are  similar  if,  and 
in  so  far  as,  they  comprehend  identical  properties.  In  respect  to  the 
remaining  properties,  they  are  different.41 

ab  represents  the  common  properties  of  a  and  b. 

a  —  ab  represents  the  peculiar  properties  of  a. 

a  +  b  —  ab  —  ab  represents  the  peculiar  properties  of  a  together  with 
the  peculiar  properties  of  b. 

It  is  evident  from  this  last  that  Lambert  does  not  wish  to  recognize  in 
his  system  the  law  a  +  a  =  a;  else  he  need  only  have  written  a  +  b  —  ab. 

If  x  and  a  are  of  the  same  genus,  then 

xy  =  ay         and         ax  =  ay  =  xy 
If  now  we  symbolize  by  a  \  b  that  part  of  a  which  is  different  from  6,42  then 

(10)  a\b  +  b\a+  ab  +  ab  =  a+b 

Also  x  —  x  a  =  ay,         or         x  =  ay  +  x  \  a 

ax  =  a8 
a  —  ax  =  ad 
a  =  ax  +  a8 

ax  —  a  —  aS  =  ay  =  xy 

41  Ibid.,  p.  10. 

42  Lambert  sometimes  uses  a  \  b  for  this,  sometimes  a  :  b. 


The  Development  of  Symbolic  Logic  23 

And  since 

ay 

X     "~~ 

7 


ax  +  a 


x  =  a  ax  +  x\a  =  x 


ax  =  a  —  a  x  =  x  —  x\a 

a\x  —  a  —  ax  x  a  =  x  —  ax 

The  fact  that  y  is  a  property  comprehended  in  x  may  be  expressed  by 
y  =  xy  or  by  y  +  x  y  =  .T.  The  manner  in  which  Lambert  deduces  the 
second  of  these  expressions  from  the  first  is  interesting.43  If  y  is  a  property 
of  x,  then  y  x  is  null.  But  by  (10), 

2xy  +  x  y  +  y  x  =  x  +  y 

Hence  in  this  case,  2xy  +  x\y  =  x  +  y 

And  since  y  =  xy,  2y  +  x\y  =  x  +  y 

Hence  y  +  x  \  y  =  x 

He  has  subtracted  y  from  both  sides,  in  the  last  step,  and  we  observe  that 
ty  —  y  =  y-  This  is  rather  characteristic  of  his  procedure;  it  follows, 
throughout,  arithmetical  analogies  which  are  quite  invalid  for  logic. 

With  the  complications  of  this  calculus,  the  reader  will  probably  be 
little  concerned.  There  is  no  general  type  of  procedure  for  elimination  or 
solution.  Formulae  of  solution  for  different  types  of  equation  are  given. 
They  are  highly  ingenious,  often  complicated,  and  of  dubious  application. 
It  is  difficult  to  judge  of  possible  applications  because  in  the  whole  course 
of  the  development,  so  far  as  outlined,  there  is  not  a  single  illustration  of  a 
solution  which  represents  logical  reasoning,  and  very  few  illustrations  of 
any  kind. 

The  shortcomings  of  this  calculus  are  fairly  obvious.  There  is  too 
much  reliance  upon  the  analogy  between  the  logical  relations  symbolized 
and  their  arithmetical  analogues.  Some  of  the  operations  are  logically 
uninterpretable,  as  for  example  the  use  of  numerical  coefficients  other  than 
0  and  1.  These  have  a  meaning  in  the  " Newtonian  formula",  but  2y  either 
has  no  meaning  or  requires  a  conventional  treatment  which  is  not  given. 
And  in  any  case,  to  subtract  y  from  both  sides  of  2y  =  x  +  y  and  get  y  =  x 
represents  no  valid  logical  operation.  Any  adequate  study  of  the  properties 
of  the  relations  employed  is  lacking,  x  =  a  +  b  is  transformed  into  a  =  x 
—  b,  regardless  of  the  fact  that  a  and  b  may  have  a  common  part  and  that 

43  Seeks  Versuche,  p.  12. 


24  A  Survey  of  Symbolic  Logic 

x  —  b  represents  the  abstraction  of  the  whole  of  b  from  x.  Suppose,  for 
example,  man  =  rational  +  animal.  Then,  by  Lambert's  procedure,  we 
should  have  also  rational  =  man  —  animal.  Since  Leibniz  had  pointed 
out  this  difficulty, — that  addition  and  subtraction  (with  exactly  these 
meanings)  are  not  true  inverses,  it  is  the  more  inexcusable  that  Lambert 
should  err  in  this. 

There  is  a  still  deeper  difficulty  here.  As  Lambert  himself  remarks,44 
no  two  concepts  are  so  completely  dissimilar  that  they  do  not  have  a  common 
part.  One  might  say  that  the  concept  "thing"  (Lambert's  word)  or  "be- 
ing" is  common  to  every  pair  of  concepts.  This  being  the  case,  [  +•  ]  and  [  —  ] 
are  never  really  inverse  operations.  Hence  the  difficulty  will  not  really 
disappear  even  in  the  case  of  ay  and  a8;  and  a  —  ay  =  a8,  a  —  ad  =  ay 
will  not  be  strictly  valid.  In  fact  this  consideration  vitiates  altogether  the 
use  of  "subtraction"  in  a  calculus  based  on  intension.  For  the  meaning 
of  a  —  b  becomes  wholly  doubtful  unless  [  — ]  be  treated  as  a  wholly  con- 
ventional inverse  of  f  +  ],  and  in  that  case  it  becomes  wholly  useless. 

The  method  by  which  Lambert  treats  the  traditional  syllogism  is  only 
remotely  connected  with  what  precedes,  and  its  value  does  not  entirely 
depend  upon  the  general  validity  of  his  calculus.  He  reconstructs  the 
whole  of  Aristotelian  logic  by  the  quantification  of  the  predicate.45 

The  proposition  "All  A  is  B"  has  two  cases: 

(1)  A  =  B,  the  case  in  which  it  has  a  universal  converse,  the  concept 
A  is  identical  with  the  concept  B. 

(2)  A  >  B,  the  case  in  which  the  converse  is  particular,  the  concept  B 
comprehended  in  the  concept  A . 

The  particular  affirmative  similarly  has  two  cases : 

(1)  A  <  B,  thejcase  in  which  the  converse  is  a  universal,  the  subject  A 
comprehended  within  the  predicate  B. 

(2)  The  case  in  which  the  converse  is  particular.     In  this  case  the 
subject  A  is  comprehended  within  a  subsumed  species  of  the  predicate  and 
the  predicate  within  a  subsumed  species  of  the  subject.     Lambert  says 
this  may  be  expressed  by  the  pair: 

mA  >  B        and        A  <  nB 

Those  who  are  more  accustomed  to  logical  relations  in  extension  must 
not  make  the  mistake  here  of  supposing  that  A  >  mA,  and  mA  <  A. 
mA  is  a  species  of  A,  and  in  intension  the  genus  is  contained  in  the  species, 

« Ibid.,  p.  12. 
•«76td.,  pp.  93  jf. 


The  Development  of  Symbolic  Logic 


25 


not  vice  versa.  Hence  mA  >  B  does  not  give  A  >  B,  as  one  might  expect 
at  first  glance.  We  see  that  Lambert  here  translates  "Some  A"  by  mA,  a 
species  comprehended  in  A,  making  the  same  assumption  which  occurs  in 
Leibniz,  that  any  subdivision  or  portion  of  a  class  is  capable  of  being  treated 
as  some  species  comprehended  under  that  class  as  its  genus. 

In  a  universal  negative  proposition — Lambert  says — the  subject  and 
predicate  each  have  peculiar  properties  by  virtue  of  whose  comprehension 
neither  is  contained  in  the  other.  But  if  the  peculiar  properties  of  the 
subject  be  taken  away,  then  what  remains  is  contained  in  the  predicate; 
and  if  the  peculiar  properties  of  the  predicate  be  taken  away,  then  what 
remains  is  contained  in  the  subject.  Thus  the  universal  negative  is  repre- 
sented by  the  pair 


m 


and 


A>B- 

n 


The  particular  negative  has  two  cases : 

(1)  When  it  has  a  universal  affirmative  converse,  i.  e.,  when  some  A 
is  not  B  but  all  B  is  A.     This  is  expressed  by 

A  <  B 

(2)  When  it  has  not  a  universal  affirmative  converse.     In  this  case  a 
subsumed  species  of  the  subject  is  contained  in  the  predicate,  and  a  sub- 
sumed species  of  the  predicate  in  the  subject. 

mA  >  B        and        A  <  nB 

Either  of  the  signs,  <  and  >,  may  be  reversed  by  transposing  the 
terms.  And  if  P  <  Q,  Q  >  P,  then  for  some  I,  P  =  IQ.  Also,  "multi- 
plication" and  "division"  are  strict  inverses.  Hence  we  can  transform 
these  expressions  as  follows: 

A  >  B  is  equivalent  to  A  =  mB 


A  <B 
mA>  B    \ 
A  <nB  ] 

A 

m 

A>B- 

n 


or        pA  =  qB 


or         —  =  — 


It  is  evident  from  these  transformations  and  from  the  prepositional  equiva- 


26  A  Survey  of  Symbolic  Logic 

lents  of  the  "inequalities"  that  the  following  is  the  full  expression  of  these 
equations : 

(1)  A  =  mB:  All  A  is  B  and  some  B  is  not  A. 

(2)  nA  =  B:  Some  A  is  not  B  and  all  B  is  A. 

(3)  mA  =  nB:  Some,  but  not  all,  A  is  B,  and  some,  but  not  all,  B  is  A. 

(4)  -  =  -  :   No  A  is  5. 

TO      w 

The  first  noticeable  defect  here  is  that  A/m  =  B/n  is  transformable  into 
nA  =  mB  and  (4)  can  mean  nothing  different  from  (3).  Lambert  has,  in 
fact,  only  four  different  propositions,  if  he  sticks  to  the  laws  of  his  calculus: 

(1)  A  =  B:  All  A  is  all  B. 

(2)  A  =  mB:  All  A  is  some  B. 

(3)  nA  =  B:  Some  A  is  all  B. 

(4)  mA  =  nB:  Some  A  is  some  B. 

These  are  the  four  forms  which  become,  in  Hamilton's  and  De  Morgan's 
treatises,  the  four  forms  of  the  affirmative.  A  little  scrutiny  will  show  that 
Lambert's  treatment  of  negatives  is  a  failure.  For  it  to  be  consistent  at 
all,  it  is  necessary  that  " fractions"  should  not  be  transformed.  But 
Lambert  constantly  makes  such  transformations,  though  he  carefully  re- 
frains from  doing  so  in  the  case  of  expressions  like  A/m  =  B/n  which  are 
supposed  to  represent  universal  negatives.  His  method  further  requires 
that  TO  and  n  should  behave  like  positive  coefficients  which  are  always 
greater  than  0  and  such  that  m  ={=  n.  This  is  unfortunate.  It  makes  it 
impossible  to  represent  a  simple  proposition  without  "entangling  alliances". 
If  he  had  taken  a  leaf  from  Leibniz's  book  and  treated  negative  propositions 
as  affirmatives  with  negative  predicates,  he  might  have  anticipated  the 
calculus  of  De  Morgan. 

In  symbolizing  syllogisms,  Lambert  always  uses  A  for  the  major  term, 
B  for  the  middle  term,  and  C  for  the  minor.  The  perfectly  general  form  of 
proposition  is : 

mA      nB 

p       q 

Hence  the  perfectly  general  syllogism  will  be : 46 

mA      nB 


Major 

-103. 
"  Ibid.,  p.  107. 


p       q 

« Ibid.,  pp.  102-103. 


The  Development  of  Symbolic  Logic  27 

nC      vB 
Minor  —  =  — 


Hn          mv 
Conclusion     —  C  =  — A 

irq  pp 

The  indeterminates  in  the  minor  are  always  represented  thus  by  Greek 
letters. 

The  conclusion  is  de   ved  from  the  premises  as  follows : 

The  major  premise  gives  B  =  — A. 

np 

The  minor  gives  B  =  —  C. 

TTV 

mq          up 

Hence  — -  A  =  —  C.   • 
np          irv 

and  therefore  —  C  =  — A. 
•n-q          pp 

The  above  being  the. general  form  of  the  syllogism,  Lambert's  scheme  of 
moods  in  the  first  figure  is  the  following:  it  coincides  with  the  traditional 
classification  only  so  far  as  indicated  by  the  use  of  the  traditional  names: 


B 

=  mA 

nB 

=  mA 

I. 

VII. 

C 

=  vB 

U,C 

=  B 

Barbara 

Lilii 

C 

=  mvB 

n/j.C 

=  mA 

B 

=  A 

II. 

C 

B 

VIII. 

B 

=  A 

7T 

U.C 

=  nB 

Canerent 

C 

A 

Magogos 

uC 

=  nA 

7T 

p 

III. 

B 

A 

IX. 

B 

=  A 

Decane 

q 
c 

P 
=  vB 

Negligo 

q 
c 

P 
=  vB 

sive 

sive 

C 

vA 

n 

vA 

Celarent 

Ferio 

_ 

=  — 

q 

P 

q 

P 

28 


A  Survey  of  Symbolic  Logic 


nB  =  A 

IV. 

C  _B 

Fideleo 

7T            p 

nC  _A 

7T             p 

X. 

Pilosos 


nB  =  A 
»C  =  B 

n/j.C  =  A 


V. 

B  = 

mA 

Gabini 

nC  = 

vB 

sive 

U,C    = 

mvA 

Darii 

nB  = 

mA 

VI. 

C  = 

B 

Hilario 

nC  = 

mA 

nB 

=  mA 

XL 

C 

=  B 

Romano 

nC 

=  mA 

nB 

=  mA 

XII. 

uC 

=  B 

Somnio 

* 

n/jiC 

=  mA 

The  difficulty  about  "division"  does  not  particularly  affect  this  scheme, 
since  it  is  only  required  that  if  one  of  the  premises  involve  "fractions", 
the  conclusion  must  also.  It  will  be  noted  that  the  mood  Hilario  is  identi- 
cal in  form  with  Romano,  and  Lilii  with  Somnio.  The  reason  for  this 
lies  in  the  fact  that  nB  =  mA  has  two  partial  meanings,  one  affirmative 
and  one  negative  (see  above).  Hilario  and  Lilii  take  the  affirmative 
interpretation,  as  their  names  indicate;  Romano  and  Somnio,  the  negative. 

Into  the  discussion  of  the  other  three  figures,  the  reader  will  probably 
not  care  to  go,  since  the  manner  of  treatment  is  substantially  the  same  as 
in  the  above. 

There  are  various  other  attempts  to  devise  a  convenient  symbolism  and 
method  for  formal  logic;48  but  these  are  of  the  same  general  type,  and 
they  meet  with  about  the  same  degree  and  kind  of  success. 

Two  brief  passages  in  which  there  is  an  anticipation  of  the  logic  of 
relatives  possess  some  interest.49  Relations,  Lambert  says,  are  "external 
attributes",  by  which  he  means  that  they  do  not  belong  to  the  object 
an  sich.  "Metaphysical"  (i.  e.,  non-logical)  relations  are  represented  by 
Greek  letters.  For  example  if  /  =  fire,  h  =  heat,  and  a  =  cause, 

f=a::h 

The  symbol   :  :  represents  a  relation  which  behaves  like  multiplication: 

48  See  in  Seeks  Versuche,  v  and  vi.  Also  fragments  "Uber  die  Vernunftlehre",  in 
Logische  und  Philosophische  Abhandlungen,  i,  xix  and  xx;  and  Anlage  zur  Architektonik, 
p.  190 ff. 

Versuche,  pp.  19,  27  ff. 


The  Development  of  Symbolic  Logic  29 

a  :  :  h  is  in  fact  what  Peirce  and  Schroder  later  called  a  "relative  product". 
Lambert  transforms  the  above  equation  into  : 

f        a 

--  =  -      Fire  is  to  heat  as  cause  to  effect. 

h 

f      Ji 

—  =  -     Fire  is  to  cause  as  heat  to  effect. 
a 

-  =  —     Heat  is  to  fire  as  effect  to  cause. 
/       a 

The  dot  here  represents  Wirkuny  (it  might  be,  Wirklichkeit,  in  consonance 
with  the  metaphysical  interpretation,  suggestive  of  Aristotle,  which  he 
gives  to  Ursache).     It  has  the  properties  of  1,  as  is  illustrated  elsewhere50 
by  the  fact  that  7°  may  be  replaced  by  this  symbol. 
Lambert  also  uses  powers  of  a  relation. 

If  a  =  (p  :  :  b,  and  b  =  <p  :  :  c, 

a  =  <p  :  :  <p  :  :  c  =  <p2  :  :  c 
And  if  a  =  <£>2  :  :  c, 

a  la 

v>  =  \- 
*c 

And  more  to  the  same  effect. 

No  use  is  made  of  this  symbolism;  indeed  it  is  difficult  to  see  how 
Lambert  could  have  used  it.  Yet  it  is  interesting  that  he  should  have  felt 
that  the  powers  of  a  relation  ought  to  be  logically  important,  and  that  he 
here  hit  upon  exactly  the  concept  by  which  the  riddles  of  "mathematical 
induction"  were  later  to  be  solved. 

Holland's  attempt  at  a  logical  calculus  is  contained  in  a  letter  to  Lam- 
bert.51 He  himself  calls  it  an  "unripe  thought",  and  in  a  letter  some  three 
years  later52  he  expresses  a  doubt  if  logic  is  really  a  purely  formal  discipline 
capable  of  mathematical  treatment.  But  this  study  is  of  particular  interest 
because  it  treats  the  logical  classes  in  extension  —  the  only  attempt  at  a 
symbolic  logic  from  the  point  of  view  of  extension  from  the  time  of  Leibniz 
to  the  treatise  of  Solly  in  1839. 

Holland  objects  to  Lambert's  method  of  representing  the  relation  of 
concepts  by  the  relation  of  lines,  one  under  the  other,  and  argues  that  the 

6J  Ibid.,  p.  21. 

51  Johan.  Lamberts  deutscher  Gelehrten  Brief  wechsel,  Brief  in,  pp.  16  ff, 

52  See  Ibid.,  Brief  xxvn,  pp.  259  ff. 


<p2  =  -        and 
c 


30  A  Survey  of  Symbolic  Logic 

relation  of  "men"  to  "mortals"  is  not  sub  but  inter.  He  is  apparently  not 
aware  that  this  means  exchanging  the  point  of  view  of  intension  for  that 
of  extension,  yet  all  his  relations  are  consistently  represented  in  extension, 
as  we  shall  see. 

(1)  If  S  represent  the  subject,  P  the  predicate;  and  p,  IT  signify  unde- 
termined variable  numbers,  S/p  =  P/TT  will  come  to:  A  part  of  <S  is  a  part 
of  P,  or  certain  of  the  S's  are  certain  of  the  P's,  or  (at  least)  an  S  is  a  P. 

This  expression  is  the  general  formula  of  all  possible  judgments,  as  is 
evident  by  the  following: 

(2)  A  member  is  either  positive  or  negative,  and  in  both  cases,  is  either 
finite  or  infinite.     We  shall  see  in  what  fashion  p  and  IT  can  be  understood. 

(3)  If  p  =  1  in  S/p,  then  is  S/p  as  many  as  all  S,  and  in  this  way  S/p 
attains  its  logical  maximum.     Since,  then,  p  cannot  become  less  than  1, 
it  can  still  less  disappear  and  consequently  cannot  become  negative. 

The  same  is  true  of  TT. 

(4)  Therefore  p  and  TT  cannot  but  be  positive  and  cannot  be  less  than  1 . 
If  p  or  TT  becomes  infinite,  the  concept  becomes  negative. 

(5)  If/  expresses  a  finite  number  >  1,  then  the  possible  forms  of  judg- 
ment are  as  follows: 

(1)  ^  =  ;      All  S  is  all  P. 

(2)  |  =  y    All  S  is  some  P. 

Now  0  expresses  negatively  what  I/*  expresses  positively.  To  say  that 
an  infinitely  small  part  of  a  curved  line  is  straight,  means  exactly :  No  part 
of  a  curved  line  is  straight. 

(3)  -  =  —    All  S  is  not  P. 

1         oo 

(4)  ^  =  ^       Some  S  is  all  P. 

J 

o         p 

(5)  -  =  -T     Some  S  is  some  P. 

j        j 

S    •    P 

(6)  -  =  —    Some  S  is  not  P. 

J        °° 


(7)  —  =  ?-      All  not-S  is  all  P. 

oo 


See  Ibid.,  Brief  iv. 


The  Development  of  Symbolic  Logic  31 

S       P 

(8)  —  =  -j      All  not-S  is  some  P. 

00       / 

S        P 

(9)  —  =  -       All  not-S  is  all  not-P. 

oo         oo 

(1),  (2),  and  (9)  Holland  says  are  universal  affirmative  propositions; 
(3),  (7),  and  (8),  universal  negatives;  (4)  and  (5),  particular  affirmatives; 
(6),  a  particular  negative. 

As  Venn  has  said,  this  notation  anticipates,  in  a  way,  the  method  of 
Boole.  If  instead  of  the  fraction  we  take  the  value  of  the  numerator 
indicated  by  it,  the  three  values  are 


where  0  <  v  <  1,  and  S/oo  =  Q-S.  But  the  differences  between  this  and 
Boole's  procedure  are  greater  than  the  resemblances.  The  fractional  form 
is  a  little  unfortunate  in  that  it  suggests  that  the  equations  may  be  cleared 
of  fractions,  and  this  would  give  results  which  are  logically  uninterpretable. 
But  Holland's  notation  can  be  made  the  basis  of  a  completely  successful 
calculus.  That  he  did  not  make  it  such,  is  apparently  due  to  the  fact  that 
he  did  not  give  the  matter  sufficient  attention  to  elaborate  the  extensional 
point  of  view. 

He  gives  the  following  examples  : 

Example  1  .  All  men  H  are  mortal  M 

All  Europeans  E  are  men  H 


7T 


M 
Ergo,        E  =  —      [All  Europeans  are  mortal] 

pir 

Example  2.  All  plants  are  organisms         P  =  - 

A 

All  plants  are  no  animals         P  =  — 

00 


0      A 
Ergo,  —  =  —    [Some  organisms  are  not  animals] 

p       oo 


32  A  Survey  of  Symbolic  Logic 

R 

Example  3.        All  men  are  rational        H  =  - 

P 

•p 
All  plants  are  not  rational        P  =  - 

00 

pH 
Ergo,  All  plants  are  no  men        P  =  — 

oo 

In  this  last  example,  Holland  has  evidently  transformed  H  =  R/p  into 
pH  =  R,  which  is  not  legitimate,  as  we  have  noted.  pH  =  R  would  be 
"Some  men  are  all  the  rational  beings".  And  the  conclusion  P  =  pHj<x> 
is  also  misinterpreted.  It  should  be,  "All  plants  are  not  some  men".  A 
correct  reading  would  have  revealed  the  invalid  operation. 

Lambert  replied  vigorously  to  this  letter,  maintaining  the  superiority 
of  the  intensional  method,  pointing  out,  correctly,  that  Holland's  calculus 
would  not  distinguish  the  merely  non-existent  from  the  impossible  or 
contradictory  (no  calculus  in  extension  can),  and  objecting  to  the  use  of 
oo  in  this  connection.  It  is  characteristic  of  their  correspondence  that  each 
pointed  out  the  logical  defects  in  the  logical  procedure  of  the  other,  and 
neither  profited  by  the  criticism. 

Castillon's  essay  toward  a  calculus  of  logic  is  contained  in  a  paper 
presented  to  the  Berlin  Academy  in  1803. 54  The  letters  S,  A,  etc.,  represent 
concepts  taken  in  intension,  M  is  an  indeterminate,  S  +  M  represents  the 
"synthesis"  of  S  and  M,  S  —  M,  the  withdrawal  or  abstraction  of  M 
from  S.  S  —  M  thus  represents  a  genus  concept  in  w^hich  S  is  subsumed, 
M  being  the  logical  "difference"  of  S  in  S  —  M .  Consonantly  S  +  M, 
symbolizing  the  addition  of  some  "further  specification"  to  S,  represents  a 
species  concept  which  contains  (in  intension)  the  concept  S. 

The  predicate  of  a  universal  affirmative  proposition  is  contained  in  the 
subject  (in  intension).  Thus  "All  S  is  A"  is  represented  by 

S  =  A  +  M 

The  universal  negative  "No  S  is  A"  is  symbolized  by 
S  =  -  A  +  M  =  (-  A)  +  M 

The  concept  S  is  something,  M,  from  which  A  is  withdrawn — is  no  A. 

Particular  propositions  are  divided  into  two  classes,  "real"  and  "il- 
lusory". A  real  particular  is  the  converse  of  a  universal  affirmative;  the 

54  "Memoire  sur  un  nouvel  algorithme  logique",  in  Memoires  de  I'Academie  des  Sciences 
de  Berlin,  1803,  Classe  de  philosophic  speculative,  pp.  1-14.  See  also  his  paper,  "Reflexions 
sur  la  Logique",  loc.  cit.,  1802. 


The  Development  of  Symbolic  Logic  33 

illusory  particular,  one  whose  converse  also  is  particular.  The  real  particu- 
lar affirmative  is 

A  =  S  -  M 

since  this  is  the  converse  of  S  =  A  +  M .  The  illusory  particular  affirmative 
is  represented  by 

S  =  A  =F  M 

Castillon's  explanation  of  this  is  that  the  illusory  particular  judgment  gives 
us  to  understand  that  some  S  alone  is  A,  or  that  S  is  got  from  A  by  ab- 
straction (S  =  A  —  M),  when  in  reality  it  is  A  which  is  drawn  from  S  by 
abstraction  (S  =  M  +  A).  Thus  this  judgment  puts  —  M  where  it  should 
put  +  M;  qne  can,  then,  indicate  it  by  S  =  A  ^  M . 

The  fact  is,  of  course,  that  "Some  S  is  A  "  indicates  nothing  about 
the  relations  of  the  concepts  S  and  A  except  that  they  are  not  incompatible. 
This  means,  in  intension,  that  if  one  or  both  be  further  specified  in  proper 
fashion,  the  results  will  coincide.  It  might  wrell  be  symbolized  by  S  +  N 
=  A  +  M.  We  suspect  that  Castillon's  choice  of  S  =  A  ^  M  is  really 
governed  by  the  consideration  that  S  =  A  +  M  may  be  supposed  to  give 
S  =  A  ^  M,  the  universal  to  give  its  subaltern,  and  that  A  =  S  —  M 
will  also  give  S  =  A  ^  M ,  that  is  to  say,  the  real  particular — which  is 
"All  A  is  S"— will  also  give  S  =  A  =F  M.  Thus  "Some  S  is  A"  may  be 
derived  both  from  "All  S  is  A"  and  from  "All  A  is  S",  which  is  a  de- 
sideratum. 

The  illusory  negative  particular  is,  correspondingly, 

S  =  -  A  =F  M 

Immediate  inference  works  out  fairly  well  in  this  symbolism. 

The  universal  affirmative  and  the  real  particular  are  converses. 

S  =  A  +  M  gives  A  =  S  —  M ,  and  vice  versa.  The  universal  negative 
is  directly  convertible. 

S  =  —A  +  M  gives  A  =  —  S  +  M ,  and  vice  versa.  The  illusory  par- 
ticular is  also  convertible. 

S  =  A  =F  M  gives  -  A  =  —  S  ^  M.  Hence  A  =  S  ^  M ,  which 
comes  back  to  S  =  A  ^  M . 

A  universal  gives  its  subaltern 

S  =  A  +  M  gives  S  =  A  =F  M ,  and 

S  =  -A  +  M  gives  S  =  -  A  =F  M. 

And  a  real  particular  gives  also  the  converse  illusory  particular,  for 

A  =  S  —  M  gives  S  =  A  +  M, 

4 


34  A  Survey  of  Symbolic  Logic 

which  gives  its  subaltern,  S  =  A  ^  M, 
which  gives  A  =  S  ^  M . 

All  the  traditional  moods  and  figures  of  the  syllogism  may  be  symbolized 
in  this  calculus,  those  which  involve  particular  propositions  being  valid 
both  for  the  real  particular  and  for  the  illusory  particular.  For  example: 

All  M  is  A  M  =  A  +  N 

AD   /Sis  M  S  =  M  +  P 

All   Sis  ,4  :.    8  =  A  +  (N  +  P) 

No  M  is  A  M  =  -  A  +  N 

All   Sis  M  S  =  M  +  P 

No   SisA  :.    S=-A  +  (N  +  P) 

All  Mia  A  M  =  A  +  N 

Some    S  is  M  S  =  M  =F  P    or    S  =  M  -  P 

.'.    Some  S  is  A  .'.     S  =  (A  +  N)  =F  P    or    S  =  (A  +  N)  -  P 

This  is  the  most  successful  attempt  at  a  calculus  of  logic  in  intension. 

The  difficulty  about  "subtraction"  in  the  XIX  Fragment  of  Leibniz, 
and  in  Lambert's  calculus,  arises  because  M  —  P  does  not  mean  "  M  but 
not  P"  or  " M  which  is  not  P  ".  If  it  mean  this,  then  [  +  ]  and  [  —  ]  are  not 
true  inverses.  If,  on  the  other  hand,  M  —  P  indicates  the  abstraction  from 
the  concept  M  of  all  that  is  involved  in  the  concept  P,  then  M  —  P  is 
difficult  or  impossible  to  interpret,  and,  in  addition,  the  idea  of  negation 
cannot  be  represented  by  [— ].  How  does  it  happen,  then,  that  Castillon's 
notation  works  out  so  well  when  he  uses  [— ]  both  for  abstraction  and  as 
the  sign  of  negation?  It  would  seem  that  his  calculus  ought  to  involve 
him  in  both  kinds  of  difficulties. 

The  answer  is  that  Castillon  has,  apparently  by  good  luck,  hit  upon  a 
method  in  wrhich  nothing  is  ever  added  to  or  subtracted  from  a  determined 
concept,  <S  or  A,  except  an  indeterminate,  M  or  N  or  P,  and  this  indeter- 
minate, just  because  it  is  indeterminate,  conceals  the  fact  that  [+  ]  and  [  — ] 
are  not  true  inverses.  And  when  the  sign  [  —  ]  appears  before  a  determinate, 
A,  it  may  serve  as  the  sign  of  negation,  because  no  difficulty  arises  from 
supposing  the  whole  of  what  is  negated  to  be  absent,  or  abstracted. 

Castillon's  calculus  is  theoretically  as  unsound  as  Lambert's,  or  more 
so  if  unsoundness  admits  of  degree.  It  is  quite  possible  that  it  was  worked 
out  empirically  and  procedures  which  give  invalid  results  avoided. 


The  Development  of  Symbolic  Logic  35 

Whoever  studies  Leibniz,  Lambert  and  Castillon  cannot  fail  to  be  con- 
vinced that  a  consistent  calculus  of  concepts  in  intension  is  either  immensely 
difficult  or,  as  Couturat  has  said,  impossible.  Its  main  difficulty  is  not 
the  one  which  troubled  Leibniz  and  which  constitutes  the  main  defect  in 
Lambert's  system — the  failure  of  [+  ]  and  [  — ]  to  behave  like  true  inverses. 
This  can  be  avoided  by  treating  negative  propositions  as  affirmatives  with 
negative  predicates,  as  Leibniz  did.  The  more  serious  difficulty  is  that  a 
calculus  of  "concepts"  is  not  a  calculus  of  things  in  actu  but  only  in  possibile, 
and  in  a  rather  loose  sense  of  the  latter  at  that.  Holland  pointed  this  out 
admirably  in  a  letter  to  Lambert.55  He  gives  the  example  according  to 
Lambert's  method. 

All  triangles  are  figures.  T  =  tF 

All  quadrangles  are  figures.     Q  =  qF 

T      0 

Whence,  F  =  -  =  -,        or        qT  =  tQ 

t        q 

and  he  then  proceeds: 56 

"In  general,  if  from  A  =  mC  and  B  =  nC  the  conclusion  nA  =  mB 
be  drawn,  the  calculus  cannot  determine  whether  the  ideas  nA  and  mB 
consist  of  contradictory  partial-ideas,  as  in  the  foregoing  example,  or  not. 
The  thing  must  be  judged  according  to  the  matter." 

This  example  also  calls  attention  to  the  fact  that  Lambert's  calculus, 
by  operations  which  he  continually  uses,  leads  to  the  fallacy  of  the  undis- 
tributed middle  term.  If  "some  A"  is  simply  some  further  specification 
of  the  concept  A,  then  this  mode  is  not  fallacious.  And  this  observation 
brings  down  the  whole  treatment  of  logic  as  a  calculus  of  concepts  in  in- 
tension like  a  house  of  cards.  The  relations  of  existent  things  cannot  be 
determined  from  the  relations  of  concepts  alone. 

The  calculus  of  Leibniz  is  more  successful  than  any  invented  by  his 
continental  successors — unless  Ploucquet's  is  an  exception.  That  the  long 
period  between  him  and  De  Morgan  and  Boole  did  not  produce  a  successful 
system  of  symbolic  logic  is  probably  due  to  the  predilection  for  this  inten- 
sional  point  of  view.  It  is  no  accident  that  the  English  were  so  quickly 
successful  after  the  initial  interest  was  aroused;  they  habitually  think  of 
logical  relations  in  extension,  and  when  they  speak  of  "intension"  it  is 
usually  clear  that  they  do  not  mean  those  relations  of  concepts  which  the 
"intension"  of  traditional  logic  signifies. 

66  Deutscher  Gekhrler  Briefwechsel,  i,  Brief  xxvu. 
66  Ibid.,  pp.  262-63. 


36  A  Survey  of  Symbolic  Logic 

The  beginning  of  thought  upon  this  subject  in  England  is  marked  by  the 
publication  of  numerous  treatises,  all  proposing  some  modification  of  the 
traditional  logic  by  quantifying  the  predicate.  As  Sir  William  Hamilton 
notes,67  the  period  from  Locke  to  1833  is  singularly  barren  of  any  real  con- 
tributions to  logic.  About  that  time,  Hamilton  himself  proposed  the 
quantification  of  the  predicate.  As  we  now  know,  this  idea  was  as  old  at 
least  as  Leibniz.  Ploucquet,  Lambert,  Holland,  and  Castillon  also  had 
quantified  the  predicate.  Both  Hamilton  and  his  student  Thomson  men- 
tion Ploucquet;  but  this  new  burst  of  logical  study  in  England  impresses 
one  as  greatly  concerned  about  its  own  innovations  and  sublimely  indifferent 
to  its  predecessors.  Hamilton  quarrelled  at  length  with  De  Morgan  to 
establish  his  priority  in  the  matter.58  This  is  the  more  surprising,  since 
George  Bentham,  in  his  Outline  of  a  New  System  of  Logic,  published  in  1827, 
had  quantified  the  predicate  and  given  the  following  table  of  propositions: 

1.  X  in  toto  =  Y  ex  parte; 

2.  X  in  toto     I  Y  ex  parte; 

3.  X  in  toto  =  Y  in  toto; 

4.  X  in  toto        Y  in  toto; 

5.  X  ex  parte  =  Y  ex  parte; 

6.  X  ex  parte        Y  ex  parte; 

7.  X  ex  parte  =  Y  in  toto; 

8.  X  ex  parte        Y  in  toto. 

(|     is  here  the  sign  of  "diversity"). 

But  Hamilton  was  certainly  the  center  and  inspirer  of  a  new  movement 
in  logic,  the  tendency  of  which  was  toward  more  precise  analysis  of  logical 
significances.  Bayne's  Essay  on  the  New  Analytic  and  Thomson's  Laws  of 
Thought  are  the  most  considerable  permanent  record  of  the  results,  but 
there  was  a  continual  fervid  discussion  of  logical  topics  in  various  peri- 
odicals; logistic  was  in  the  air. 

This  movement  produced  nothing  directly  which  belongs  to  the  history  of 
symbolic  logic.  Hamilton's  rather  cumbersome  notation  is  not  made  the 
basis  of  operations,  but  is  essentially  only  an  abbreviation  of  language. 
Solly's  scheme  of  representing  syllogisms  was  superior  as  a  calculus.  But 

67  See  Discussions  on  Philosophy,  pp.  119  jf. 

68  This  controversy,  begun  in  1846,  was  continued  for  many  years  (see  various  articles 
in  the  London  Athenaeum,  from  1860  to  1867).     It  was  concluded  in  the  pages  of  the  Con- 
temporary Review,  1873. 


The  Development  of  Symbolic  Logic  3  7 

this  movement  accomplished  two  things  for  symbolic  logic:  it  emphasized 
in  fact — though  not  always  in  name — the  point  of  view  of  extension,  and 
it  aroused  interest  in  the  problem  of  a  newer  and  more  precise  logic.  These 
may  seem  small,  but  whoever  studies  the  history  of  logic  in  this  period 
will  easily  convince  himself  that  without  these  things,  symbolic  logic  might 
never  have  been  revived.  Without  Hamilton,  we  might  not  have  ha'd 
Boole.  The  record  of  symbolic  logic  on  the  continent  is  a  record  of  failure, 
in  England,  a  record  of  success.  The  continental  students  habitually 
emphasized  intension;  the  English,  extension. 

IV.     DE  MOPGAN 

De  Morgan59  is  known  to  most  students  of  symbolic  logic  only  through 
the  theorem  which  bears  his  name.  But  he  made  other  contributions  of 
permanent  value — the  idea  of  the  "universe  of  discourse",60  the  discovery 
of  certain  new  types  of  propositions,  and  a  beginning  of  the  logic  of  rela- 
tions. Also,  his  originality  in  the  invention  of  new  logical  forms,  his  ready 
wit,  his  pat  illustrations,  and  the  clarity  and  liveliness  of  his  writing  did 
yeoman  service  in  breaking  down  the  prejudice  against  the  introduction 
of  "mathematical"  methods  in  logic.  His  important  writings  on  logic 
are  comprised  in  the  Formal  Logic,  the  Syllabus  of  a  Proposed  System  of 
Logic,  and  a  series  of  articles  in  the  Transactions  of  the  Cambridge  Philo- 
sophical Society.61 

69  Augustus  De  Morgan  (1806-78),  A.B.  (Cambridge,  1827),  Professor  of  Mathematics 
in  the  University  of  London  1828-31,  reappointed  1835;  writer  of  numerous  mathematical 
treatises  which  are  characterized  by  exceptional  accuracy,  originality  and  clearness.  Per- 
haps the  most  valuable  of  these  is  "Foundations  of  Algebra"  (Camb.  Phil.  Trans.,  vn, 
vni) ;  the  best  known,  the  Budget  of  Paradoxes.  For  a  list  of  his  papers,  see  the  Royal 
Society  Catalogue.  For  many  years  an  active  member  of  the  Cambridge  Philosophical 
Society  and  the  Royal  Astronomical  Society.  Father  of  William  F.  De  Morgan,  the  novelist 
and  poet.  For  a  brief  biography,  see  Monthly  Notices  of  the  Royal  Astronomical  Society, 
xii,  112. 

60  The  idea  is  introduced  with  these  words:    "Let  us  take  a  pair  of  contrary  names, 
as  man  and  not-man.     It  is  plain  that  between  them  they  represent  everything,  imaginable 
or  real,  in  the  universe.     But  the  contraries  of  common  language  embrace,  not  the  whole 
universe,  but  some  one  general  idea.     Thus,  of  men,  Briton  and  alien  are  contraries: 
every  man  must  be  one  of  the  two,  no  man  can  be  both.  .  .  .  The  same  may  be  said  of 
integer  and  fraction  among  numbers,  peer  and  commoner  among  subjects  of  a  realm, 
male  and  female  among  animals,  and  so  on.     In  order  to  express  this,  let  us  say  that  the 
whole  idea  under  consideration  is  the  universe  (meaning  merely  the  whole  of  which  we  are 
considering  parts)  and  let  names  which  have  nothing  in  common,  but  which  between  them 
contain  the  whole  of  the  idea  under  consideration,  be  called  contraries  in,  or  with  respect  to, 
that  universe."     (Formal  Logic,  p.  37;  see  also  Camb.  Phil.  Trans.,  vm,  380.) 

61  Formal  Logic:   or,  The  Calculus  of  Inference,  Necessary  and  Probable,  1847.     Here- 
after to  be  cited  as  F.  L. 


A  Survey  of  Symbolic  Logic 

Although  the  work  of  De  Morgan  is  strictly  contemporary  with  that  of 
Boole,  his  methods  and  symbolism  ally  him  rather  more  with  his  prede- 
cessors than  with  Boole  and  those  who  follow.  Like  Hamilton,  he  is  bent 
upon  improving  the  traditional  Aristotelian  logic.  His  first  step  in  this 
direction  is  to  enlarge  the  number  of  typical  propositions  by  considering 
all  the  combinations  and  distributions  of  two  terms,  X  and  Y,  and  their 
negatives.  It  is  a  feature  of  De  Morgan's  notation  that  the  distribution  of 
each  term,62  and  the  quality — affirmative  or  negative — of  the  proposition 
are  indicated,  these  being  sufficient  to  determine  completely  the  type  of 
the  proposition. 

That  a  term  A"  is  distributed  is  indicated  by  writing  half  a  parenthesis 
Jbefore  or  after  it,  with  the  horns  turned  toward  the  letter,  thus:  X),  or  (X. 
-An  undistributed  term  is  marked  by  turning  the  half-parenthesis  the  other 
~«vay,  thus:  X(,  or  }X.  X))Y,  for  example,  indicates  the  proposition  in 
which  the  subject,  X,  is  distributed  and  the  predicate,  Y,  is  undistributed, 
that  is,  "All  X  is  7".  XQY  indicates  a  proposition  with  both  terms  un- 
distributed, that  is,  "Some  X  is  Y".63  The  negative  of  a  term,  X,  is  indi- 
cated by  .T;  of  Y  by  y,  etc.  A  negative  proposition  is  indicated  by  a  dot 
placed  between  the  parenthetical  curves;  thus  "Some  JT  is  not  Y"  will 
be  X(-(Y.6*  Two  dots,  or  none,  indicates  an  affirmative  proposition. 

All  the  different  forms  of  proposition  which  De  Morgan  uses  can  be 
generated  from  two  types,  the  universal,  "All  .  .  .  is  .  .  .,"  and  the 
particular,  "Some  .  .  .is  .  .  .,"  by  using  the  four  terms,  X  and  its  nega- 
tive, x,  Y  and  y.  For  the  universals  we  have: 

Syllabus  of  a  Proposed  System  of  Logic,  1860.     Hereafter  to  be  cited  as  Syll. 

Five  papers  (the  first  not  numbered;  various  titles)  in  Camb.  Phil.  Trans.,  vm,  ix,  x. 

The  articles  contain  the  most  valuable  material,  but  they  are  ill-arranged  and  inter- 
spersed with  inapposite  discussion.  Accordingly,  the  best  way  to  study  De  Morgan  is  to 
get  these  articles  and  the  Formal  Logic,  note  in  a  general  way  the  contents  of  each,  and 
then  use  the  Syllabus  as  a  point  of  departure  for  each  item  in  which  one  is  interested. 

62  He  does  not  speak  of  "distribution"  but  of  terms  which  are  "universally  spoken  of" 
or  "particularly  spoken  of  ",  or  of  the  "quantity"  of  a  term. 

63  This  is  the  notation  of  Syll.  and  of  the  articles,  after  the  first,  in  Camb.  Phil.  Trans. 
For  a  table  comparing  the  different  symbolisms  which  he  used,  see  Camb.  Phil.  Trans., 
ix,  91. 

64  It  is  sometimes  hard  to  determine  by  the  conventional  criteria  whether  De  Morgan's 
propositions  should  be  classed  as  affirmative  or  negative.     He  gives  the  following  ingenious 
rule  for  distinguishing  them  (Syll.,  p.  13):    "Let  a  proposition  be  affirmative  which  is  true 
of  X  and  X,  false  of  X  and  not-.X"  or  x;  negative,  which  is  true  of  X  and  x,  false  of  X  and  X. 
Thus  'Every  X  is  Y'  is  affirmative:   'Every  X  is  X'  is  true;   'Every  X  is  x'  is  false.     But 
'Some  things  are  neither  X's  nor  Y's'  is  also  affirmative,  though  in  the  form  of  a  denial: 
'Some  things  are  neither  X's  nor  X's'  is  true,  though  superfluous  in  expression;    'Some 
things  are  neither  X's  nor  x's'  is  false." 


The  Development  of  Symbolic  Logic  39 

(1)  Z))F  All  X  is  F. 

(2)  x)}y  All  not-Z  is  not- Y. 

(3)  X}}y  All  Z  is  not-F. 

(4)  z))F  All  not-Z  is  F. 

and  for  particulars  we  have : 

(5)  Z()F  Some  X  is  F. 

(6)  x()y  Some  not-Z  is  not-F. 

(7)  XQy  Some  Z  is  not-F. 

(8)  z()F  Somenot-Zis  F. 

The  rule  for  transforming  a  proposition  into  other  equivalent  forms  may 
be  stated  as  follows:  Change  the  distribution  of  either  term— that  is,  turn 
its  parenthetic  curve  the  other  way, — change  that  term  into  its  negative, 
and  change  the  quality  of  the  proposition.  That  this  rule  is  valid  will 
appear  if  we  remember  that  "two  negatives  make  an  affirmative",  and  note 
that  we  introduce  one  negative  by  changing  the  term,  another  by  changing 
the  quality  of  the  proposition.  That  the  distribution  of  the  altered  term 
should  be  changed  follows  from  the  fact  that  whatever  proposition  distrib- 
utes a  term  leaves  the  negative  of  that  term  undistributed,  and  whatever 
proposition  leaves  a  term  undistributed  distributes  the  negative  of  that 
term.  Using  this  rule  of  transformation,  we  get  the  following  table  of 
equivalents  for  our  eight  propositions: 

(a)  (b)  (c)  (d) 

(1)  Z))F  =  Z).(</   =x((y    =z(-)F 

(2)  x}}y    =  x}.(Y   =Z((F  =  Z(-)2/ 

(3)  X}}y   =  X}-(Y  =  x((Y    =  x(-}y 

(4)  .r))F    =  xY(y     =X((y   =  Z(-)F 

(5)  XQY  =  X(.(y   =x}(y     =  z)-)F 

(6)  x()y     =x(.(Y    =Z)(F  =  Z)-)2/ 

(7)  X(}y   =X(-(Y  =  x)(Y    =  x}-}y 

(8)  xQY   =x(-(y    =X}(y   =Z)-)F 

It  will  be  observed  that  in  each  line  there  is  one  proposition  with  both 
terms  positive,  Z  and  F.  Selecting  these,  we  have  the  eight  different  types 
of  propositions: 


40  A  Survey  of  Symbolic  Logic 

(la)  AT))y  All  X  is  7. 

(2c)  X((Y  Some  X  is  all  7;  or,  All  Y  is  AT. 

(36)  X)-(Y  No  AT  is  7. 

(4d)  A'(-)y  Everything  is  either  X  or  y.     (See  below.) 

(5a)  XQY  Some  X  is  y. 

(6c)  X)(Y  Some  things  are  neither  X  nor  y.     (See  below.) 

(76)  X(-(Y  Some  X  is  not  y. 

(8d)  X)  •)  y  All  AT  is  not  some  y;  or,  Some  Y  is  not  X. 

Since  the  quantity  of  each  term  is  indicated,  any  one  of  these  propositions 
may  be  read  or  written  backwards  —  that  is,  with  y  subject  and  X  predicate 
—provided  the  distribution  of  terms  is  preserved.  (4d)  and  (6c)  are  diffi- 
cult to  understand.  We  might  attempt  to  read  X(-}Y  "Some  A'  is  not 
some  y",  but  we  hardly  get  from  that  the  difference  between  X(-}Y  and 
X(-(Y,  "Some  X  is  not  (any)  Y".  Also,  A"(-)y  is  equivalent  to  uni- 
versals,  and  the  reading,  "Some  X  is  not  some  Y",  would  make  it  par- 
ticular. X(-)Y  is  equivalent  to  a:))  y,  "All  not-Z  is  y",  and  to  x)-(y, 
"No  not-A"  is  not-y".  The  only  equivalent  of  these  with  the  terms 
X  and  y  is,  "Everything  (in  the  universe  of  discourse)  is  either  A'  or  y 
(or  both)".  (6c),  X)(Y,  we  should  be  likely  to  read  "All  X  is  all  Y",  or 
"A"  and  y  are  equivalent";,  but  this  would  be  an  error,65  since  its  equivalents 
are  particular  propositions.  (6a),  x()y,  is  "Some  not-  A"  is  not-y". 
The  equivalent  of  this  in  terms  of  X  and  y  is  plainly,  "Some  things  are 
neither  X  nor  y". 

Contradictories66  of  propositions  in  line  (1)  will  be  found  in  line  (7); 
of  those  in  line  (2),  in  line  (8);  of  line  (3),  in  line  (5);  of  line  (4),  in  line  (6). 
We  give  those  with  both  terms  positive  : 

(la)  Z))y  contradicts  (76)  X(-(Y 

(2c)  X((Y  "  (So7)  A»}7 

(36)  X)-(Y  "  (5a)  XQY 

(.)Y  "  (6c)  X)(Y 


65  An  error  into  which  it  might  seem  that  De  Morgan  himself  has  fallen.     See  e.  g., 
Syll,  p.  25,  and  Camb.  Phil.  Trans.,  ix,  98,  where  he  translates  X)(Y  by  "All  X  is  all  Y", 
or  "Any  one  X  is  any  one  Y  ".     But  this  belongs  to  another  interpretation,  the  "cumular", 
which  requires  X  and  Y  to  be  singular,  and  not-X  and  not-F  will  then  have  common 
members.     However,  as  we  shall  note  later,  there  is  a  real  difficulty. 

66  De  Morgan  calls  contradictory  propositions  "contraries"  (See  F.  L.,  p.  60;  Syll., 
p.  11),  just  as  he  calls  terms  which  are  negatives  of  one  another  "contraries". 


The  Development  of  Symbolic  Logic  41 

Thus  the  rule  is  that  two  propositions  having  the  same  terms  contradict 
one  another  when  one  is  affirmative,  the  other  negative,  and  the  distribution 
of  terms  is  exactly  opposite  in  the  two  cases. 

The  rule  for  transforming  propositions  which  has  been  stated  and 
exemplified,  together  with  the  observation  that  any  symbolized  proposition 
may  be  read  or  written  backwards,  provided  the  distribution  of  the  terms 
be  preserved,  gives  us  the  principles  for  the  immediate  inference  of  uni- 
versals  from  universals,  particulars  from  particulars.  For  the  rest,  we  have 
the  rule,  "Each  universal  affirms  the  particulars  of  the  same  quality".67 

For  syllogistic  reasoning,  the  test  of  validity  and  rule  of  inference  are 
as  follows: 6& 

"There  is  inference:  1.  When  both  the  premises  are  universal;  2.  When, 
one  premise  only  being  particular,  the  middle  term  has  different  quantities 
in  the  two  premises. 

"  The  conclusion  is  found  by  erasing  the  middle  term  and  its  quantities 
[parenthetic  curves]."  This  rule  of  inference  is  stated  for  the  special 
arrangement  of  the  syllogism  in  wrhich  the  minor  premise  is  put  first,  and 
the  minor  term  first  in  the  premise,  the  major  term  being  the  last  in  the 
second  premise.  Since  any  proposition  may  be  written  backward,  this 
arrangement  can  always  be  made.  According  to  the  rule,  X)}Y,  "All  X 
is  F",  and  F)-(Z,  "No  Y  is  Z",  give  X)-(Z,  "No  X  is  Z".  X)-(F,  "No 
X  is  F",  and  Y(-  (Z,  "Some  Y  is  not  Z",  give  X)  •  •  (Z,  or  X)  (Z,  which  is 
"Some  things  are  neither  X  nor  Z. "  The  reader  may,  by  inventing  other 
examples,  satisfy  himself  that  the  rule  given  is  sufficient  for  all  syllogistic 
reasoning,  with  any  of  De  Morgan's  eight  forms  of  propositions. 

De  Morgan  also  invents  certain  compound  propositions  which  give  com- 
pound syllogisms  in  a  fashion  somewhat  analogous  to  the  preceding: 69 

"1.  Z)0)Forboth.Y))FandZ)-)F     All  X's  and  some  things  be- 
sides are  F's. 

2.  X\    For  both  Z))F  and  Z((F          All  X's  are   F's,  and  all   F's 

are  X's. 

3.  X(O(Y  or  both  X((Y  and  X(-(Y     Among  X's  are  all  the  F's  and 

some  things  besides. 

4.  X} O (For  both  X}-(Y  and  X)(Y     Nothing  both  X  and    F  and 

some  things  neither. 

67  Syll,  p.  16. 
«*Syll.,  p.  19. 
69  Sytt.,  p.  22. 


42  A  Survey  of  Symbolic  Logic 

5.-X\-  For  both  A>(Fand  X(>)Y  Nothing  both  X  and  Y  and 

everything  one  or  the  other. 

6.  X(O)Y  or  both  X(-)Y  and  XQY  Everything  either  X  or  Y  and 

some  things  both." 

Each  of  these  propositions  may,  with  due  regard  for  the  meaning  of  the 
sign  O,  be  read  or  written  backward,  just  as  the  simple  propositions.  The 
rule  of  transformation  into  other  equivalent  forms  is  slightly  different: 
Change  the  quantity,  or  distribution,  of  any  term  and  replace  that  term 
by  its  negative.  We  are  not  required,  as  with  the  simple  propositions,  to 
change  at  the  same  time  the  quality  of  the  proposition.  This  difference 
is  due  to  the  manner  in  which  the  propositions  are  compounded. 

The  rules  for  mediate,  or  "syllogistic",  inference  for  these  compound 
propositions  are  as  follows : 70 

"If  any  two  be  joined,  each  of  which  is  [of  the  form  of]  1,  3,  4,  or  6, 
with  the  middle  term  of  different  quantities,  these  premises  yield  a  con- 
clusion of  the  same  kind,  obtained 'by  erasing  the  symbols  of  the  middle 
term  and  one  of  the  symbols  [Q]-  Thus  X}O(Y(O}Z  gives  X)O)Z:  or 
if  nothing  be  both  X  and  Y  and  some  things  neither,  and  if  everything  be 
either  Y  or  Z  and  some  things  both,  it  follows  that  all  X  and  two  lots  of 
other  things  are  Z's. 

"In  any  one  of  these  syllogisms,  it  follows  that  may  be  written  for 
)O)  or  )O(  in  one  place,  without  any  alteration  of  the  conclusion,  except 
reducing  the  two  lots  to  one.  But  if  this  be  done  in  both  places,  the  con- 
clusion is  reduced  to  j  or  •  ,  and  both  lots  disappear.  Let  the  reader 
examine  for  himself  the  cases  in  which  one  of  the  premises  is  cut  down  to  a 
simple  universal. 

"The  following  exercises  will  exemplify  what  precedes.  Letters  written 
under  one  another  are  names  of  the  same  object.  Here  is  a  universe  of  12 
instances  of  which  3  are  X's  and  the  remainder  P's;  5  are  F's  and  the 
remainder  Q's;  7  are  Z's  and  the  remainder  R's. 

XXX  PP  PP  PPPPP 

YYY  YY  qq  qqqqq 

Z  Z  Z     Z  Z    Z  Z    RRRRR 
We  can  thus  verify  the  eight  complex  syllogisms 

X)omo)Z        P(0)Y)0)Z        P(Q(Q(o)Z        P(O(Q(O(R 
P(0)Y)0(R        X)0)Y)0(R       X)0(Q(0(R 


The  Development  of  Symbolic  Logic  43 

In  every  case  it  will  be  seen  that  the  two  lots  in  the  middle  form  the  quantity 
of  the  particular  proposition  of  the  conclusion." 

In  so  much  of  his  work  as  we  have  thus  far  reviewed,  De  Morgan  is  still 
too  much  tied  to  his  starting  point  in  Aristotelian  logic.  He  somewhat 
simplifies  traditional  methods  and  makes  new  generalizations  which  include 
old  rules,  but  it  is  still  distinctly  the  old  logic.  He  does  not  question  the 
inference  from  universals  to  particulars  nor  observe  the  problems  there 
involved.71  He  does  not  seek  a  method  by  which  any  number  of  terms 
may  be  dealt  with  but  accepts  the  limitation  to  the  traditional  two.  And 
his  symbolism  has  several  defects.  The  dot  introduced  between  the 
parenthetic  curves  is  not  the  sign  of  negation,  so  as  to  make  it  possible  to 
read  (•)  as,  "It  is  false  that  ()".  The  negative  of  ()  is  )•(,  so  that  this 
simplest  of  all  relations  of  propositions  is  represented  by  a  complex  trans- 
formation applicable  only  when  no  more  than  two  terms  are  involved  in  the 
prepositional  relation.  Also,  there  are  two  distinct  senses  in  which  a 
term  in  a  proposition  may  be  distributed  or  "mentioned  universally",  and 
De  Morgan,  following  the  scholastic  tradition,  fails  to  distinguish  them  and 
symbolizes  both  the  same  way.  This  is  the  secret  of  the  difficulty  in  reading 
X)(Y,  which  looks  like  "All  X  is  all  Y",  and  really  is  "Some  things  are 
neither  X  nor  Y  ".72  Mathematical  symbols  are  introduced  but  without  any 
corresponding  mathematical  operations.  The  sign  of  equality  is  used  both 
for  the  symmetrical  relation  of  equivalent  propositions  and  for  the  un- 
symmetrical  relation  of  premises  to  their  conclusion.73 

His  investigation  of  the  logic  of  relations,  however,  is  more  successful, 
and  he  laid  the  foundation  for  later  researches  in  that  field.  This  topic 
is  suggested  to  him  by  consideration  of  the  formal  and  material  elements 
in  logic.  He  says: 74 

71  But  he  does  make  the  assumption  upon  which  all  inference   (in  extension)  of  a 
particular  from  a  universal  is  necessarily  based :  the  assumption  that  a  class  denoted  by  a 
simple  term  has  members.     He  says  (F.  L.,  pp.  110),  "Existence  as  objects,  or  existence  as 
ideas,  is  tacitly  claimed  for  the  terms  of  every  syllogism". 

72  A  universal  affirmative  distributes  its  subject  in  the  sense  that  it  indicates  the  class 
to  which  every  member  of  the  subject  belongs,  i.  e.,  the  class  denoted  by  the  predicate. 
Similarly,  the  universal  negative,  No  X  is  Y,  indicates  that  every  X  is  not-Y",  every  F  is 
not-X.     No  particular  proposition  distributes  a  term  in  that  sense.     The  particular  nega- 
tive tells  us  only  that  the  predicate  is  excluded  from  some  unspecified  portion  of  the  class 
denoted  by  the  subject.     X)(Y  distributes  X  and  Y  in  this  sense  only.     Comparison  with 
its  equivalents  shows  us  that  it  can  tell  us,  of  X,  only  that  it  is  excluded  from  some  un- 
specified portion  of  not-F;  and  of  Y,  only  that  it  is  excluded  from  some  unspecified  portion 
of  not-X.     We  cannot  infer  that  X  is  wholly  included  in  Y,  or  Y  in  X,  or  get  any  other 
relation  of  inclusion  out  of  it. 

73  In  one  passage  (Carafe.  Phil.  Trans.,  x,  183)  he  suggests  that  the  relation  of  two 
premises  to  their  conclusion  should  be  symbolized  by  A  B  <  C. 

74  Camb.  Phil.  Trans.,  x,  177,  footnote. 


44  A  Survey  of  Symbolic  Logic 

"Is  there  any  consequence  without /orm?  Is  not  consequence  an  action 
of  the  machinery?  Is  not  logic  the  science  of  the  action  of  the  machinery? 
Consequence  is  always  an  act  of  the  mind :  on  every  consequence  logic  ought 
to  ask,  What  kind  of  act?  What  is  the  act,  as  distinguished  from  the  acted 
on,  and  from  any  inessential  concomitants  of  the  action  ?  For  these  are  of 
the  form,  as  distinguished  from  the  matter. 

"...  The  copula  performs  certain  functions;  it  is  competent  to  those 
functions  .  .  .  because  it  has  certain  properties,  which  are  sufficient  to 
validate  its  use.  .  .  .  The  word  'is,'  which  identifies,  does  not  do  its  work 
because  it  identifies,  except  insofar  as  identification  is  a  transitive  and 
convertible  motion:  'A  is  that  which  is  B'  means  'A  is  B';  and  'A  is  B' 
means  'B  is  A'.  Hence  every  transitive  and  convertible  relation  is  as  fit 
to  validate  the  syllogism  as  the  copula  'is',  and  by  the  same  proof  in  each 
case.  Some  forms  are  valid  when  the  relation  is  only  transitive  and  not 
convertible;  as  in  'give'.  Thus  if  X — Y  represent  X  and  Y  connected 
by  a  transitive  copula,  Camestres  in  the  second  figure  is  valid,  as  in 

Every  Z—Y,     No  X—  Y,    therefore     No  X— Z. 

...  In  the  following  chain  of  propositions,  there  is  exclusion  of  matter, 
form  being  preserved  at  every  step : 

Hypothesis 
(Positively  true)     Every  man  is  animal 

Every  man  is  Y  Y  has  existence. 

Every  X  is  Y  X  has  existence. 

Every  X — Y  -  is  a  transitive  relation. 

a  of  A" — Y  a  is  a  fraction  <  or  =  1. 

(Probability  j8)        a  of  X — 7  )8  is  a  fraction  <  or  =  1. 

The  last  is  nearly  the  purely  formal  judgment,  with  not  a  single  material 
point  about  it,  except  the  transitiveness  of  the  copula.75 

"...  I  hold  the  supreme  form  of  the  syllogism  of  one  middle  term  to 
be  as  follows:  There  is  the  probability  a  that  X  is  in  the  relation  L  to  7; 
there  is  the  probability  /3  that  Y  is  in  the  relation  M  to  Z;  whence  there  is 
the  probability  a/3  that  X  is  in  the  relation  L  of  M  to  Z.76 

"...  The  copula  of  cause  and  effect,  of  motive  and  action,  of  all  which 
post  hoc  is  of  the  form  and  propter  hoc  (perhaps)  of  the  matter,  will  one  day 
be  carefully  considered  in  a  more  complete  system  of  logic."  77 

75  Ibid.,  pp.  177-78. 

76  Ibid.,  p.  339. 

77  Ibid.,  pp.  179-80. 


The  Development  of  Symbolic  Logic  45 

De  Morgan  is  thus  led  to  a  study  of  the  categories  of  exact  thinking  in 
general,  and  to  consideration  of  the  types  and  properties  of  relations. 
His  division  of  categories  into  logico-mathematical,  logico-physical,  logico- 
metaphysical,  and  logico-contraphysical,78  is  inauspicious,  and  nothing 
much  comes  of  it.  But  in  connection  with  this,  and  an  attempt  to  rebuild 
logic  in  the  light  of  it,  he  propounds  the  well-known  theorem:  "The  con- 
trary [negative]  of  an  aggregate  [logical  sum]  is  the  compound  [logical 
product]  of  the  contraries  of  the  aggregants:  the  contrary  of  a  compound 
is  the  aggregate  of  the  contraries  of  the  components."  79 

For  the  logic  of  relations,  X,  Y,  and  Z  will  represent  the  class  names; 
L,  M,  N,  relations.  X  .  .  LY  will  signify  that  X  is  some  one  of  the  objects 
of  thought  which  stand  to  Y  in  the  relation  L,  or  is  one  of  the  L's  of  F.80 
X  .  L  Y  will  signify  that  X  is  not  any  one  of  the  L's  of  Y.  X  .  .  (LM)  Y  or 
X  .  .  LM  Y  will  express  the  fact  that  X  is  one  of  the  L's  of  one  of  the  M' s 
of  Y,  or  that  X  has  the  relation  L  to  some  Z  which  has  the  relation  M  to  Y. 
X  .  LM  Y  will  mean  that  X  is  not  an  L  of  any  M  of  Y. 

It  should  be  noted  that  the  union  of  the  two  relations  L  and  M  is  what 
we  should  call  today  their  "relative  product" ;  that  is,  X  .  .LY  and  Y  .  .  MZ 
together  give  X  .  .  LM  Z,  but  X  .  .  LY  and  X  .  .  M  Y  do  not  give  X  .  .  LM  Y. 
If  L  is  the  relation  "brother  of"  and  M  is  the  relation  "aunt  of",  X  .  .  LM  Y 
will  mean  "  X  is  a  brother  of  an  aunt  of  F".  (Do  not  say  hastily,  "  X  is 
uncle  of  F".  "Brother  of  an  aunt"  is  not  equivalent  to  "uncle"  since 
some  uncles  have  no  sisters.)  L,  or  M,  written  by  itself,  will  represent 
that  which  has  the  relation  L,  or  M,  that  is,  a  brother,  or  an  aunt,  and  LY 
stands  for  any  X  which  has  the  relation  L  to  Y,  that  is,  a  brother  of  I7.81 

In  order  to  reduce  ordinary  syllogisms  to  the  form  in  which  the  copula 
has  that  abstractness  which  he  seeks,  that  is,  to  the  form  in  which  the 
copula  may  be  any  relation,  or  any  relation  of  a  certain  type,  it  is  necessary 
to  introduce  symbols  of  quantity.  Accordingly  LM  *  is  to  signify  an  L  of 
every  M,  that  is,  something  which  has  the  relation  L  to  every  member  of 
the  class  M  (say,  a  lover  of  every  man).  L*M  is  to  indicate  an  L  of  none 
but  M's  (a  lover  of  none  but  men).  The  mark  of  quantity,  *  or  *,  always 

78  See  ibid.,  p.  190. 

79  Ibid.,  p.  208.     See  also  Syll.,  p.  41.     Pp.  39-60  of  Syll.  present  in  summary  the  ideas 
of  the  paper,  "On  the  Syllogism,  No.  3,  and  on  Logic  in  General.'' 

80  Camb.  Phil.  Trans.,  x,  341.     We  follow  the  order  of  the  paper  from  this  point  on. 

81 1  tried  at  first  to  make  De  Morgan's  symbolism  more  readily  intelligible  by  intro- 
ducing the  current  equivalents  of  his  characters.  But  his  systematic  ambiguities,  such 
as  the  use  of  the  same  letter  for  the  relation  and  for  that  which  has  the  relation,  made 
this  impossible.  For  typographical  reasons,  I  use  the  asterisk  where  he  has  a  small  accent. 


46  A  Survey  of  Symbolic  Logic 

goes  with  the  letter  which  precedes  it,  but  L*M  is  read  as  if  {*]  modified 
the  letter  which  follows.  To  obviate  this  difficulty,  De  Morgan  suggests 
that  L*M  be  read,  "An  every-!/  of  M;  an  L  of  M  in  every  way  in  which 
it  is  an  L,"  but  we  shall  stick  to  the  simpler  reading,  "An  L  of  none  but 
M's". 

LM*X  means  an  L  of  every  M  of  X:  L*MX,  an  L  of  none  but  M's  of  X: 
L*M*,  an  L  of  every  M  and  of  none  but  M's:  LMX*,  an  L  of  an  M  of 
every  X,  and  so  on. 

Two  more  symbols  are  needed.  The  converse  of  L  is  symbolized  by  L~l. 
If  L  is  "lover  of",  L~l  is  "beloved  of";  if  L  is  "aunt",  L~l  is  "niece  or 
nephew  ".  The  contrary  (or  as  we  should  say,  the  negative)  of  L  is  symbol- 
ized by  1;  the  contrary  of  M  by  m. 

In  terms  of  these  relations,  the  following  theorems  can  be  stated : 

(1)  Contraries  of  converses  are  themselves  contraries. 

(2)  Converses  of  contraries  are  contraries. 

(3)  The  contrary  of  the  converse  is  the  converse  of  the  contrary. 

(4)  If  the  relation  L  be  contained  in,  or  imply,  the  relation  M ,  then  (a)  the 
converse  of  L,  L~l,  is  contained  in  the  converse  of  M,  M~l\   and  (6)  the 
contrary  of  M,  m,  is  contained  in  the  contrary  of  L,  I. 

For  example,  if  "parent  of  "  is  contained  in  "ancestor  of",  (a)  "child  of" 
is  contained  in  "descendent  of",  and  (6)  "not  ancestor  of"  is  contained  in 
"not  parent  of". 

(5)  The  conversion  of  a  compound  relation  is  accomplished  by  converting 
both  components  and  inverting  their  order;  thus,  (LM)~l  =  M~1L~*. 

If  X  be  teacher  of  the  child  of  Y,  Y  is  parent  of  the  pupil  of  X. 

When  a  sign  of  quantity  is  involved  in  the  conversion  of  a  compound 
relation,  the  sign  of  quantity  changes  its  place  on  the  letter;  thus,  (LM*)~l 
=  M*-1!.-1. 

If  X  be  teacher  of  every  child  of  Y,  Y  is  parent  of  none  but  pupils  of  X. 

(6)  When,  in  a  compound  relation,  there  is  a  sign  of  quantity,  if  each 
component  be  changed  into  its  contrary,  and  the  sign  of  quantity  be  shifted 
from  one  component  to  the  other  and  its  position  on  the  letter  changed, 
the  resulting  relation  is  equivalent  to  the  original;   thus  LM *  =  l*m  and 
L*M  =  1m*. 

A  lover  of  every  man  is  a  non-lover  of  none  but  non-men;  and  a  lover 
of  none  but  men  is  a  non-lover  of  every  non-man. 


The  Development  of  Symbolic  Logic  47 

(7)  When  a  compound  relation  involves  no  sign  of  quantity,  its  contrary 
is  found  by  taking  the  contrary  of  either  component  and  giving  quantity 
to  the  other.     The  contrary  of  LM  is  IM*  or  L*m. 

"Not  (lover  of  a  man)"  is  "non-lover  of  every  man"  or  "lover  of  none 
but  non-men";  and  there  are  two  equivalents,  by  (6). 

But  if  there  be  a  sign  of  quantity  in  one  component,  the  contrary  is 
taken  by  dropping  that  sign  and  taking  the  contrary  of  the  other  component. 
The  contrary  of  LM  *  is  IM ;  of  L*M  is  Lm. 

"Not  (lover  of  every  man)"  is  "non-lover  of  a  man";  and  "not  (lover 
of  none  but  men)"  is  "lover  of  a  non-man". 

So  far  as  they  do  not  involve  quantifications,  these  theorems  are  familiar 
to  us  today,  though  it  seems  not  generally  known  that  they  are  due  to 
De  Morgan.  The  following  table  contains  all  of  them: 

Converse  of  Contrary 
Combination  Converse  Contrary  Contrary  of  Converse 

LM  M~lL~l  IM*  or  L*M       M*^l~l  or  m'lL~l* 

LM*orl*m       M*~lL~l  or  m^l'1*  IM  M^H 

L*M  or  Im*       M~lL~l*  or  m*~ll~l  Lm  m^L'1 

The  sense  in  which  one  relation  is  said  to  be  "contained  in"  or  to 
"imply"  another  should  be  noted:  L  is  contained  in  M  in  case  every  X 
which  has  the  relation  L  to  any  Y  has  also  the  relation  M  to  that  F.  This 
must  not  be  confused  with  the  relation  of  class  inclusion  between  two  rela- 
tive terms.  Every  grandfather  is  also  a  father,  the  class  of  grandfathers  is 
contained  in  the  class  of  fathers,  but  "grandfather  of"  is  not  contained  in 
"father  of",  because  the  grandfather  of  Y  is  not  also  the  father  of  Y.  The 
relation  "grandfather  of"  is  contained  in  "ancestor  of",  since  the  grand- 
father of  F  is  also  the  ancestor  of  F.  But  De  Morgan  appropriately  uses 
the  same  symbol  for  the  relation  " L  contained  in  M "  that  he  uses  for  "All 
L  is  M ",  where  L  and  M  are  class  terms,  that  is,  L))M. 

In  terms  of  this  relation  of  relations,  the  following  theorems  can  be 
stated : 

(8)  If  L))M,  then  the  contrary  of  M  is  contained  in  the  contrary  of  L,— 
that  is,  L))M  gives  ra))/. 

Applying  this  theorem  to  compound  relations,  we  have: 

(8')    LM))N  gives  n))lM*  and  n))L*m. 
(8"}  If  LM))N,  then  L~ln)}m  and  nlf-1))/. 

Proof:  If  LM))N,  then  n)}lM*.     Whence  nM-l))lM*M~1.     But  an  /  of 


48  A  Survey  of  Symbolic  Logic 

every  M  of  an  M~l  of  Z  must  be  an  /  of  Z.  Hence  nM~l))l.  Again;  if 
LM))N,  then  n))Z,*m.  Whence  L-ln~))L-lL*m.  But  whatever  has  the 
relation  converse-of-Z  to  an  L  of  none  but  m's  must  be  itself  an  m.  Hence 
I-ln))m. 

De  Morgan  calls  this  "theorem  K  "  from  its  use  in  Baroko  and  Bokardo. 

(9)  If  LM  =  N,  then  L))NM~l  and  M))L~1N. 

Proof:  If  LM  =  N,  then  LMM~l  =  tf  Jf-1  and  L~1LM  =  L~1N.  Now 
for  any  X,  MM~1X  and  L~1LX  are  classes  which  contain  X',  hence  the 
theorem. 

We  do  not  have  L  =  NM~l  and  M  =  L~1N,  because  it  is  not  generally 
true  that  MM~1X  =  X  and  L~1LX  =  X.  For  example,  the  child  of  the 
parent  of  X  may  not  be  X  but  A"'s  brother  :  but  the  class  "  children  of  the 
parent  of  X"  will  contain  X.  The  relation  MM~l  or  M~1M  will  not  always 
cancel  out.  MM'1  and  M~1M  are  always  symmetrical  relations  ;  if  XMM~l  Y 
then  YMM~1X.  If  X  is  child  of  a  parent  of  Y,  then  Y  is  child  of  a  parent 
of  A'.  But  MM-1  and  M^M  are  not  exclusively  reflexive.  XMM~1X  does 
not  always  hold.  If  we  know  that  a  child  of  the  parent  of  X  is  a  celebrated 
linguist  we  may  not  hastily  assume  that  X  is  the  linguist  in  question. 

With  reference  to  transitive  relations,  we  may  quote  :  82 

"A  relation  is  transitive  when  a  relative  of  a  relative  is  a  relative  of 
the  same  kind;  as  symbolized  in  LL)}L,  whence  ZZZ))ZZ))Z;  and  so  on. 

"A  transitive  relation  has  a  transitive  converse,  but  not  necessarily  a 
transitive  contrary:  for  L~lL~l  is  the  converse  of  LL,  so  that  ZZ))Z  gives 
Z^Zr^Zr1.  From  these,  by  contraposition,  and  also  by  theorem  K  and 
its  contrapositions,  we  obtain  the  following  results  : 

L  is  contained  in  LL-1*,  Ul~l,  l~ll*,  L*~1L 
L-1   ..........  L*L~l,  II-1*,  l*~ll,  L~1L* 

I  .............  IL*,L*1 

I-1  ...........  Z*-1/-1,  HI-1* 

LL  ...........  L 


L^l,  IL-1  ......  I 

LI-1,  l^L  ......  H 

"I  omit  demonstration,  but  to  prevent  any  doubt  about  correctness  of 
printing,  I  subjoin  instances  in  words:  L  signifies  ancestor  and  L-1  descendent. 

82  Camb.  Phil.  Trans.,  x,  346.  For  this  discussion  of  transitive  relations,  De  Morgan 
treats  all  reciprocal  relations,  such  as  XLL~1Y,  as  also  reflexive,  though  not  necessarily 
exclusively  reflexive. 


The  Development  of  Symbolic  Logic  49 

"An- ancestor  is  always  an  ancestor  of  all  descendents,  a  non-ancestor 
of  none  but  non-descendents,  a  nbn-descendent  of  all  non-ancestors,  and  a 
descendent  of  none  but  ancestors.  A  descendent  is  always  an  ancestor  of 
none  but  descendents,  a  non-ancestor  of  all  non-descendents,  a  non-descend- 
ent  of  none  but  non-ancestors,  and  a  descendent  of  all  ancestors.  A  non- 
ancestor  is  always  a  non-ancestor  of  all  ancestors,  and  an  ancestor  of  none 
but  non-ancestors.  A  non-descendent  is  a  descendent  of  none  but  non- 
descendents,  and  a  non-descendent  of  all  descendents.  Among  non- 
ancestors  are  contained  all  descendents  of  non-ancestors,  and  all  non- 
ancestors  of  descendents.  Among  non-descendents  are  contained  all 
ancestors  of  non-descendents,  and  all  non-descendents  of  ancestors." 

In  terms  of  the  general  relation,  L,  or  M,  representing  any  relation,  the 
syllogisms  of  traditional  logic  may  be  tabulated  as  follows:  K 


1 

2 

3 

4 

X 

..LY 

X  . 

LY 

X  . 

.LY 

X 

.LY 

I 

Y 

..LZ 

Y  . 

.MZ 

Y  . 

MZ 

Y 

.MZ 

X 

.  .  LMZ 

X  . 

.IMZ 

X  . 

.LmZ 

X 

.  .  ImZ 

X 

.LY 

X  . 

.LY 

X  . 

.LY 

X 

.LY 

II 

Z 

..MY 

Z  . 

.MY 

Z  . 

MY 

Z 

.MY 

X 

.  .  IM~1Z 

X  . 

.  LM~1Z 

X  . 

.  Lm~lZ 

X 

.  .  lm-U 

Y 

..LX 

Y  . 

LX 

Y  . 

.LX 

Y 

.LX 

III 

Y 

.MZ 

Y  . 

.MZ 

Y  . 

.MZ 

Y 

.MZ 

X 

.  .  L~lmZ 

X  . 

.  i-wz 

X  . 

.  L-WZ 

X 

.  .  l~lmZ 

Y 

.LX 

Y  . 

.LX 

Y  . 

LX 

Y 

..LX 

IV 

Z 

.MY 

Z  . 

MY 

Z  . 

.MY 

Z 

..MY 

X 

4-lmrlZ 

X  . 

.  L~lm-lZ 

X  . 

.  1~1M~1Z 

X 

.  L-[M-] 

The  Roman  numerals  here  indicate  the  traditional  figures.  All  the  con- 
clusions are  given  in  the  affirmative  form;  but  for  each  affirmative  con- 
clusion, there  are  two  negative  conclusions,  got  by  negating  the  relation  and 
replacing  it  by  one  or  the  other  of  its  contraries.  Thus  X  .  .  LMZ  gives 
X  .  IM*Z  and  X  .  L*mZ;  X  .  .  IM~1Z  gives  X  .  LM~l*Z  and  X  .  l*m~lZ, 
and  so  on  for  each  of  the  others. 

83  Ibid.,  p.  350. 
5 


50  A  Survey  of  Symbolic  Logic 

When  the  copula  of  all  three  propositions  is  limited  to  the  same  transitive 
relation,  L,  or  its  converse,  the  table  of  syllogisms  will  be : 84 

X..LY       X.LY  X..LY 

I        Y..LZ        Y..L~1Z        Y.L~1Z 

X . .  LZ        X.LZ  X.  L-^Z 


X  ..LY 
Z  .LY 
X  .  L~1Z 


X  . 

LY 

X  . 

.LY 

II 

z. 

.LY 

Z. 

.L-*Y 

X  . 

LZ 

X  . 

.LZ 

Y  . 

.LX 

Y  . 

LX 

III 

Y  . 

LZ 

Y  . 

.LZ 

X  . 

LZ 

X  . 

L~1Z 

Y  . 

.LX 

IV 

Z  . 

L~1Y 

X  . 

LZ 

Y  ..LX 
Y  .  .  L~1Z 
X  .  .  L~1Z 


Y.LX  Y..LX 

Z..L~1Y        Z..LY 
X.L^Z          X..L~1Z" 

Here,  again,  in  the  logic  of  relations,  De  Morgan  would  very  likely  have 
done  better  if  he  had  left  the  traditional  syllogism  to  shift  for  itself.  The 
introduction  of  quantifications  and  the  systematic  ambiguity  of  L,  M, 
etc.,  which  are  used  to  indicate  both  the  relation  and  that  which  has  the 
relation,  hurry  him  into  complications  before  the  simple  analysis  of  rela- 
tions, and  types  of  relations,  is  ready  for  them.  This  logic  of  relations  was 
destined  to  find  its  importance  in  the  logistic  of  mathematics,  not  in  any 
applications  to,  or  modifications  of,  Aristotelian  logic.  And  these  compli- 
cations of  De  Morgan's,  due  largely  to  his  following  the  clues  of  formal  logic, 
had  to  be  discarded  later,  after  Peirce  discovered  the  connection  between 
Boole's  algebra  and  relation  theory.  The  logic  of  relative  terms  has  been 
reintroduced  by  the  wrork  of  Frege  and  Peano,  and  more  especially  of 
Whitehead  and  Russell,  in  the  logistic  development  of  mathematics.  But 
it  is  there  separated — and  has  to  be  separated — from  the  simpler  analysis 
of  the  relations  themselves.  Nevertheless,  it  should  always  be  remembered 
that  it  was  De  Morgan  who  laid  the  foundation;  and  if  some  part  of  his 
work  had  to  be  discarded,  still  his  contribution  was  indispensable  and  of 
permanent  value.  In  concluding  his  paper  on  relations,  he  justly  remarks: 85 

84  Ibid.,  p.  354. 

85  Ibid.,  p.  358. 


The  Development  of  Symbolic  Logic  51 

"And  here  the  general  idea  of  relation  emerges,  and  for  the  first  time 
in  the  history  of  knowledge,  the  notions  of  relation  and  relation  of  relation 
are  symbolized.  And  here  again  is  seen  the  scale  of  graduation  of  forms, 
the  manner  in  which  what  is  difference  of  form  at  one  step  of  the  ascent  is 
difference  of  matter  at  the  next.  But  the  relation  of  algebra  to  the  higher 
developments  of  logic  is  a  subject  of  far  too  great  extent  to  be  treated  here. 
It  will  hereafter  be  acknowledged  that,  though  the  geometer  did  not  think 
it  necessary  to  throw  his  ever-recurring  principiwn  et  exemplum  into  imita- 
tion of  Omnis  homo  est  animal,  Sortes  est  homo,  etc.,  yet  the  algebraist  was 
living  in  the  higher  atmosphere  of  syllogism,  the  unceasing  composition  of 
relation,  before  it  was  admitted  that  such  an  atmosphere  existed."  86 

V.     BOOLE 

The  beginning  from  which  symbolic  logic  has  had  a  continuous  develop- 
ment is  that  made  by  George  Boole.87  His  significant  and  vital  contribution 
was  the  introduction,  in  a  fashion  more  general  and  systematic  than  before, 
of  mathematical  operations.  Indeed  Boole  allows  operations  which  have 
no  direct  logical  interpretation,  and  is  obviously  more  at  home  in  mathe- 
matics than  in  logic.  It  is  probably  the  great  advantage  of  Boole's  wrork 
that  he  either  neglected  or  was  ignorant  of  those  refinements  of  logical 
theory  which  hampered  his  predecessors.  The  precise  mathematical 
development  of  logic  needed  to  make  its  own  conventions  and  interpreta- 
tions; and  this  could  not  be  done  without  sweeping  aside  the  accumulated 
traditions  of  the  non-symbolic  Aristotelian  logic.  As  we  shall  see,  all  the 
nice  problems  of  intension  and  extension,  of  the  existential  import  of  uni- 
versals  and  particulars,  of  empty  classes,  and  so  on,  return  later  and  demand 
consideration.  It  is  well  that,  with  Boole,  they  are  given  a  vacation  long 
enough  to  get  the  subject  started  in  terms  of  a  simple  and  general  procedure. 

Boole's  first  book,  The  Mathematical  Analysis  of  Logic,  being  an  Essay 
toward  a  Calculus  of  Deductive  Reasoning,  was  published  in  1847,  on  the 

88 1  omit,  with  some  misgivings,  any  account  of  De  Morgan's  contributions  to  prob- 
ability theory  as  applied  to  questions  of  authority  and  judgment.  (See  Syll,  pp.  67-72; 
F.  L.,  Chap,  ix,  x;  and  Camb.  Phil.  Trans.,  vm,  384-87,  and  393-405.)  His  work  on  this 
topic  is  less  closely  connected  with  symbolic  logic  than  was  Boole's.  The  allied  subject  of 
the  "numerically  definite  syllogism"  (see  Syll.,  pp.  27-30;  F.  L.,  Chap,  vm;  and  Camb. 
Phil.  Trans.,  x,  *355-*358)  is  also  omitted. 

87  George  Boole  (1815-1864)  appointed  Professor  of  Mathematics  in  Queen's  College, 
Cork,  1849;  LL.D.  (Dublin,  1852),  F.R.S.  (1857),  D.C.L.  (Oxford,  1859).  For  a  biographi- 
cal sketch,  by  Harley,  see  Brit.  Quart.  Rev.,  XLIV  (1866),  141-81.  See  also  Proc.  Roy. 
Soc.,  xv  (1867),  vi-xi. 


52  ,   A  Survey  of  Symbolic  Logic 

same  day  as  De  Morgan 's  Formal  Logic.68  The  next  year,  his  article,  "The 
Calculus  of  Logic,"  appeared  in  the  Cambridge  Mathematical  Journal.  This 
article  summarizes  very  briefly  and  clearly  the  important  innovations  pro- 
posed by  Boole.  But  the  authoritative  statement  of  his  system  is  found 
in  An  Investigation  of  the  Laws  of  Thought,  on  which  are  founded  the  Mathe- 
matical Theories  of  Logic  and  Probability,  published  in  1854.89 

Boole's  algebra,  unlike  the  systems  of  his  predecessors,  is  based  squarely 
upon  the  relations  of  extension.  The  three  fundamental  ideas  upon  which 
his  method  depends  are:  (1)  the  conception  of  "elective  symbols";  (2)  the 
laws  of  thought  expressed  as  rules  for  operations  upon  these  symbols;  (3) 
the  observation  that  these  rules  of  operation  are  the  same  which  would 
hold  for  an  algebra  of  the  numbers  0  and  I.90 

For  reasons  which  will  appear  shortly,  the  "universe  of  conceivable 
objects"  is  represented  by  1.  All  other  classes  or  aggregates  are  supposed 
to  be  formed  from  this  by  selection  or  limitation.  This  operation  of  electing, 
in  1,  all  the  A"'s,  is  represented  by  l-x  or  x;  the  operation  of  electing  all 
the  F's  is  similarly  represented  by  l-y  or  y,  and  so  on.  Since  Boole  does 
not  distinguish  between  this  operation  of  election  represented  by  x,  and 
the  result  of  performing  that  operation — an  ambiguity  common  in  mathe- 
matics—a- becomes,  in  practice,  the  symbol  for  the  class  of  all  the  X's. 
Thus  x,  y,  z,  etc.,  representing  ambiguously  operations  of  election  or  classes, 
are  the  variables  of  the  algebra.  Boole  speaks  of  them  as  "  elective  symbols" 
to  distinguish  them  from  coefficients. 

This  operation  of  election  suggests  arithmetical  multiplication:  the 
'suggestion  becomes  stronger  when  we  note  that  it  is  not  confined  to  1. 
1  •  x  •  y  or  xy  will  represent  the  operation  of  electing,  first,  all  the  X 's  in  the 
"universe",  and  from  this  class  by  a  second  operation,  all  the  F's.  The 
result  of  these  two  operations  will  be  the  class  whose  members  are  both 
X's  and  Y's.  Thus  xy  is  the  class  of  the  common  members  of  x  and  y; 
xyz,  the  class  of  those  things  which  belong  at  once  to  x,  to  y,  and  to  2, 
and  so  on.  And  for  any  x,  1  -x  =  x. 

The  operation  of  "aggregating  parts  into  a  whole"  is  represented  by  +  . 
x  +  y  symbolizes  the  class  formed  by  combining  the  two  distinct  classes, 
x  and  y.  It  is  a  distinctive  feature  of  Boole's  algebra  that  x  and  y  in  x  +  y 
must  have  no  common  members.  The  relation  may  be  read,  "that  which 

88  See  De  Morgan's  note  to  the  article  "On  Propositions  Numerically  Definite",  Camb. 
Phil.  Trans.,  xi  (1871),  396. 

89  London,  Walton  and  Maberly. 

90  This  principle  appears  for  the  first  time  in  the  Laws  of  Thought.     See  pp.  37-38. 
Work  hereafter  cited  as  L.  of  T. 


The  Development  of  Symbolic  Logic  53 

is  either  x  or  y  but  not  both".  Although  Boole  does  not  remark  it,  x  +  y 
cannot  be  as  completely  analogous  to  the  corresponding  operation  of 
ordinary  algebra  as  xy  is  to  the  ordinary  algebraic  product.  In  numerical 
algebras  a  number  may  be  added  to  itself:  but  since  Boole  conceives  the 
terms  of  any  logical  sum  to  be  "  quite  distinct",91  mutually  exclusive  classes, 
x  +  x  cannot  have  a  meaning  in  his  system.  As  we  shall  see,  this  is  very 
awkward,  because  such  expressions  still  occur  in  his  algebra  and  have  to  be 
dealt  with  by  troublesome  devices. 

But  making  the  relation  x  +  y  completely  disjunctive  has  one  advantage 
— it  makes  possible  the  inverse  relation  of  "subtraction".  The  "separa- 
tion of  a  part,  x,  from  a  whole,  y",  is  represented  by  y  —  x.  If  x  +  z  =  y, 
then  since  x  and  z  have  nothing  in  common,  y  —  x  =  z  and  y  —  z  =  x. 
Hence  [+  ]  and  [  — ]  are  strict  inverses. 

x  +  y,  then,  symbolizes  the  class  of  those  things  which  are  either  members 
of  x  or  members  of  y,  but  not  of  both,  x-y  or  xy  symbolizes  the  class  of 
those  things  which  are  both  members  of  x  and  members  of  y.  x  —  y  repre- 
sents the  class  of  the  members  of  x  which  are  not  members  of  y — the  x's 
except  the  y's.  [  =  ]  represents  the  relation  of  two  classes  which  have  the 
same  members,  i.  e.,  have  the  same  extension.  These  are  the  fundamental 
relations  of  the  algebra. 

The  entity  (1  —  x)  is  of  especial  importance.  This  represents  the 
universe  except  the  x's,  or  all  things  which  are  not  x's.  It  is,  then,  the 
supplement  or  negative  of  x. 

With  the  use  of  this  symbolism  for  the  negative  of  a  class,  the  sum  of  two 
classes,  x  and  y,  which  have  members  in  common,  can  be  represented  by 

xy  +  x(l  -  y}  +  (1  -  x)y. 

The  first  term  of  this  sum  is  the  class  which  are  both  x's  and  y's',  the  second, 
those  which  are  x's  but  not  y's;  the  third,  those  which  are  y's  but  not  x's. 
Thus  the  three  terms  represent  classes  which  are  all  mutually  exclusive, 
and  the  sum  satisfies  the  meaning  of  +  .  In  a  similar  fashion,  x  +  y  may 
be  expanded  to 

x(l  -  y}  +  (l  -  x)y. 

Consideration  of  the  laws  of  thought  and  of  the  meaning  of  these  sym- 
bols will  show  us  that  the  following  principles  hold : 

(1)  xy  =  yx  What  is  both  x  and  y  is  both  y  and  x. 

(2)  x  +  y  =  y  +  x  What  is  either  x  or  y  is  either  y  or  x. 
91  See  L.  of  T.,  pp.  32-33. 


54  A  Survey  of  Symbolic  Logic 

(3)  z(x  +  y)  =  zx  +  zy  That  which  is  both  z  and  (either  x  or  y) 

is  either  both  z  and  x  or  both  z  and  y. 

(4)  z(x  —  y)  =  zx  —  zy  That  which  is  both  2  and  (x  but  not  y) 

is  both  z  and  a;  but  not  both  z  and  y. 

(5)  If  ar  =  y,  then  22  =  zi/ 

2  +  x  =  z  +  y 
x  —  z  =  y  —  z 
(fyx-y=-y  +  x 

This  last  is  an  arbitrary  convention:   the  first  half  of  the  expression  gives 
the  meaning  of  the  last  half. 

It  is  a  peculiarity  of  "logical  symbols"  that  if  the  operation  x,  upon  1, 
be  repeated,  the  result  is  not  altered  by  the  repetition : 

l-x  =  l-x-x  =  1-X'X-x. . ,.        Hence  we  have : 
(7)  .-c2  =  x 

Boole  calls  this  the  "index  law".92 

All  these  laws,  except  (7),  hold  for  numerical  algebra.  It  may  be 
noted  that,  in  logic,  "If  a:  =  y,  then  zx  —  zy"  is  not  reversible.  At  first 
glance,  this  may  seem  to  be  another  difference  between  numerical  algebra 
and  the  system  in  question.  But  "If  zx  =  zy,  then  x  =  y"  does  not  hold 
in  numerical  algebra  when  2=0.  Law  (7)  is,  then,  the  distinguishing 
principle  of  this  algebra.  The  only  finite  numbers  for  which  it  holds  are 
0  and  1.  All  the  above  laws  hold  for  an  algebra  of  the  numbers  0  and  1.  With 
this  observation,  Boole  adopts  the  entire  procedure  of  ordinary  algebra, 
modified  by  the  law  x2  =  x,  introduces  numerical  coefficients  other  than  0 
and  1,  and  makes  use,  on  occasion,  of  the  operation  of  division,  of  the 
properties  of  functions,  and  of  any  algebraic  transformations  which  happen 
to  serve  his  purpose.93 

This  borrowing  of  algebraic  operations  which  often  have  no  logical 
interpretation  is  at  first  confusing  to  the  student  of  logic;  and  commen- 
tators have  seemed  to  smile  indulgently  upon  it.  An  example  will  help: 
the  derivation  of  the  "law  of  contradiction"  or,  as  Boole  calls  it,  the  "law 
of  duality",  from  the  "index  law".94 

92  In  Mathematical  Analysis  of  Logic  he  gives  it  also  in  the  form  xn  =  x,  but  in  L.  of  T. 
he  avoids  this,  probably  because  the  factors  of  xn  —  x  (e.  g.,  x3  —  x)  are  not  always  logically 
interpretable. 

93  This  procedure  characterizes  L.  of  T.     Only  0  and  1,  and  the  fractions  which  can 
be  formed  from  them  appear  in  Math.  An.  of  Logic,  and  the  use  of  division  and  of  fractional 
coefficients  is  not  successfully  explained  in  that  book. 

94  L.  of  T.,  p.  49. 


The  Development  of  Symbolic  Logic  55 

Since  x2  =  x,     x  —  x2  =  0. 

Hence,  factoring,  x(l  —  x)  =  0. 

This  transformation  hardly  represents  any  process  of  logical  deduction. 
Whoever  says  "What  is  both  x  and  x,  x2,  is  equivalent  to  x;  therefore  what 
is  both  x  and  not-z  is  nothing"  may  well  be  asked  for  the  steps  of  his  reason- 
ing. Nor  should  we  be  satisfied  if  he  reply  by  interpreting  in  logical  terms 
the  intermediate  expression,  x  —  x2  =  0. 

Nevertheless,  this  apparently  arbitrary  way  of  using  uninterpretable 
algebraic  processes  is  thoroughly  sound.  Boole's  algebra  may  be  viewed 
as  an  abstract  mathematical  system,  generated  by  the  laws  we  have  noted, 
which  has  two  interpretations.  On  the  one  hand,  the  "logical"  or  "elec- 
tive "  symbols  may  be  interpreted  as  variables  whose  value  is  either  numeri- 
cal 0  or  numerical  1,  although  numerical  coefficients  other  than  0  and  1  are 
admissible,  provided  it  be  remembered  that  such  coefficients  do  not  obey 
the  "index  law"  which  holds  for  "elective"  symbols.  All  the  usual  alge- 
braic transformations  will  have  an  interpretation  in  these  terms.  On  the 
other  hand,  the  "logical"  or  "elective"  symbols  may  be  interpreted  as 
logical  classes.  For  this  interpretation,  some  of  the  algebraical  processes 
of  the  system  and  some  resultant  expressions  will  not  be  expressible  in  terms 
of  logic.  But  whenever  they  are  interpretable,  they  will  be  valid  conse- 
quences of  the  premises,  and  even  when  they  are  not  interpretable,  any 
further  results,  derived  from  them,  which  are  interpretable,  will  also  be 
valid  consequences  of  the  premises. 

It  must  be  admitted  that  Boole  himself  does  not  observe  the  proprieties 
of  his  procedure.  His  consistent  course  would  have  been  to  develop  this  al- 
gebra without  reference  to  logical  meanings,  and  then  to  discuss  in  a  thorough 
fashion  the  interpretation,  and  the  limits  of  that  interpretation,  for  logical 
classes.  By  such  a  method,  he  would  have  avoided,  for  example,  the 
difficulty  about  x  +  x.  We  should  have  x  +  x  =  2x,  the  interpretation  of 
which  for  the  numbers  0  and  1  is  obvious,  and  its  interpretation  for  logical 
classes  would  depend  upon  certain  conventions  which  Boole  made  and 
which  will  be  explained  shortly.  The  point  is  that  the  two  interpretations 
should  be  kept  separate,  although  the  processes  of  the  system  need  not  be 
limited  by  the  narrower  interpretation — that  for  logical  classes.  Instead 
of  making  this  separation  of  the  abstract  algebra  and  its  two  interpretations, 
Boole  takes  himself  to  be  developing  a  calculus  of  logic;  he  observes  that 
its  "axioms"  are  identical  with  those  of  an  algebra  of  the  numbers  0  and  1; 95 
95  L.  of  T.,  pp.  37-38. 


56  A  Survey  of  Symbolic  Logic 

hence  he  applies  the  whole  machinery  of  that  algebra,  yet  arbitrarily  rejects 
from  it  any  expressions  which  are  not  finally  interpretable  in  terms  of  logical 
relations.  The  retaining  of  non-interpretable  expressions  which  can  be 
transformed  into  interpretable  expressions  he  compares  to  "the  employ- 
ment of  the  uninterpretable  symbol  V— 1  in  the  intermediate  processes 
of  trigonometry."96  It  would  be  a  pretty  piece  of  research  to  take  Boole's 
algebra,  find  independent  postulates  for  it  (his  laws  are  entirely  insufficient 
as  a  basis  for  the  operations  he  uses),  complete  it,  and  systematically  investi- 
gate its  interpretations. 

But  neglecting  these  problems  of  method,  the  expression  of  the  simple 
logical  relations  in  Boole's  symbolism  will  now  be  entirely  clear.  Classes 
wril)  be  represented  by  x,  y,  z,  etc.;  their  negatives,  by  (1  —  x),  (1  —  y), 
etc.  That  which  is  both  x  and  y  will  be  xy;  that  which  is  x  but  not  y  will 
be  a:(l  —  y),  etc.  That  which  is  x  or  y  but  not  both,  will  be  x  +  y,  or 
x(l  —  y}  +  (1  —  x)y.  That  which  is  x  or  y  or  both  wrill  be  x  +  (1  —  x)y — 
i.  e.,  that  which  is  x  or  not  x  but  y—or 

xy  +  x(l  -  y)  +  (1  -  x)y 

— that  wrhich  is  both  x  and  y  or  x  but  not  y  or  y  but  not  x.  1  represents  the 
"universe"  or  "everything".  The  logical  significance  of  0  is  determined 
by  the  fact  that,  for  any  y,  Oy  =  0 :  the  only  class  \vhich  remains  unaltered 
by  any  operation  of  electing  from  it  whatever  is  the  class  "nothing". 

Since  Boole's  algebra  is  the  basis  of  the  classic  algebra  of  logic — which 
is  the  topic  of  the  next  chapter — it  will  be  unnecessary  to  comment  upon 
those  parts  of  Boole's  procedure  which  were  taken  over  into  the  classic 
algebra.  These  will  be  clear  to  any  who  understand  the  algebra  of  logic 
in  its  current  form  or  who  acquaint  themselves  with  the  content  of  Chapter 
II.  We  shall,  then,  turn  our  attention  chiefly  to  those  parts  of  his  method 
which  are  peculiar  to  him. 

Boole  does  not  symbolize  the  relation  "x  is  included  in  ?/".  Conse- 
quently the  only  copula  by  which  the  relation  of  terms  in  a  proposition  can 
be  represented  is  the  relation  =.  And  since  all  relations  are  taken  in 
extension,  x  =  y  symbolizes  the  fact  that  x  and  y  are  classes  with  identical 
membership.  Propositions  must  be  represented  by  equations  in  which 
something  is  put  =  0  or  =  1,  or  else  the  predicate  must  be  quantified. 
Boole  uses  both  methods,  but  mainly  relies  upon  quantification  of  the 
predicate.  This  involves  an  awkward  procedure,  though  one  which  still 
survives — the  introduction  of  a  symbol  v  or  w,  to  represent  an  indefinite 

91 L.  of  T.,  p.  69. 


The  Development  of  Symbolic  Logic  57 

class  and  symbolize  "Some".  Thus  "All  x  is  (some)  y"  is  represented  by 
x  =  vy:  "Some  x  is  (some)  y",  by  wx  =  vy.  If  v,  or  w,  were  here  "the 
indefinite  class"  or  "any  class",  this  method  would  be  less  objectionable. 
But  in  such  cases  v,  or  w,  must  be  very  definitely  specified:  it  must  be  a 
class  "indefinite  in  all  respects  but  this,  that  it  contains  some  members  of 
the  class  to  whose  expression  it  is  prefixed".97  The  universal  affirmative 
can  also  be  expressed,  without  this  symbol  for  the  indeterminate,  as  x(l  —  y} 
=  0;  "All  x  is  y"  means  "That  which  is  x  but  not  y  is  nothing".  Negative 
propositions  are  treated  as  affirmative  propositions  with  a  negative  predi- 
cate. So  the  four  typical  propositions  of  traditional  logic  are  expressed  as 
follows:98 

All  x  is  y:  x  =  vy,  or,  x(l  —  y}  =  0. 

Xo  x  is  y:  x  =  v(l  —  y),  or  xy  =  0. 

Some  x  is  y:  vx  =  w(\  —  y),  or,  v  =  xy. 

Some  x  is  not  y:  vx  =  iv(l  —  y},  or,  v  =  x(l  —  y}. 

Each  of  these  has  various  other  equivalents  which  may  be  readily  deter- 
mined by  the  laws  of  the  algebra. 

To  reason  by  the  aid  of  this  symbolism,  one  has  only  to  express  his 
premises  explicitly  in  the  proper  manner  and  then  operate  upon  the  resultant 
equation  according  to  the  laws  of  the  algebra.  Or,  as  Boole  more  explicitly 
puts  it,  valid  reasoning  requires:  99 

"  1st,  That  a  fixed  interpretation  be  assigned  to  the  symbols  employed 
in  the  expression  of  the  data;  and  that  the  laws  of  the  combination  of  these 
symbols  be  correctly  determined  from  that  interpretation. 

"  2nd,  That  the  formal  processes  of  solution  or  demonstration  be  con- 
ducted throughout  in  obedience  to  all  the  laws  determined  as  above,  with- 
out regard  to  the  question  of  the  interpretation  of  the  particular  results 
obtained. 

"3rd,  That  the  final  result  be  interpretable  in  form,  and  that  it  be 
actually  interpreted  in  accordance  with  that  system  of  interpretation  which 
has  been  employed  in  the  expression  of  the  data." 

As  we  shall  see,  Boole's  methods'  of  solution  sometimes  involve  an 
uninterpretable  stage,  sometimes  not,  but  there  is  provided  a  machinery  by 

97  L.  of  T.,  p.  63.     This  translation  of  the  arbitrary  v  by  "Some"  is  unwarranted,  and 
the  above  statement  is  inconsistent  with  Boole's  later  treatment  of  the  arbitrary  coefficient. 
There  is  no  reason  why  such  an  arbitrary  coefficient  may  not  be  null. 

98  See  Math.  An.  of  Logic,  pp.  21-22;  L.  of  T.,  Chap.  iv. 

99  L.  of  T.,  p.  68. 


58  A  Survey  of  Symbolic  Logic 

which  any  equation  may  be  reduced  to  a  form  which  is  interpretable.  To 
comprehend  this  we  must  first  understand  the  process  known  as  the  develop- 
ment of  a  function.  With  regard  to  this,  we  can  be  brief,  because  Boole's 
method  of  development  belongs  also  to  the  classic  algebra  and  is  essentially 
the  process  explained  in  the  next  chapter.100 

Any  expression  in  the  algebra  which  involves  x  or  (1  —  x)  may  be 
called  a  function  of  x.  A  function  of  x  is  said  to  be  developed  when  it  has 
the  form  Ax  +  B(l  —  x}.  It  is  here  required  that  x  be  a  "logical  symbol", 
susceptible  only  of  the  values  0  and  1.  But  the  coefficients,  A  and  B,  are 
not  so  limited:  A,  or  B,  may  be  such  a  "logical  symbol"  which  obeys  the 
"law  of  duality",  or  it  may  be  some  number  other  than  0  or  1,  or  involve 
such  a  number.  If  the  function,  as  given,  does  not  have  the  form  Ax 
+  B(l  —  x},  it  may  be  put  into  that  form  by  observing  certain  interesting 
laws  which  govern  coefficients. 

Let  /(.r)  =  Ax  +  B(l  -  x} 

Then  /(I)  =  A-l  +  B(l  -  1)  =  A 

And  /(O)  =  A-0  +  B(l  -  0)  =  B 

Hence  f(x)  =  /(I)  -x  +/(0)  •  (1  -  z) 

Thus  if  /(*)  =  ^  , 

2  —  x 

fm      1  +  1  -2-      wm..  1  +  0  -1 

-  2^~l  "  2^o  -  2 

Hence  f(x)  =  2x+  -  (1  —  x) 

£ 

A  developed  function  of  two  variables,  x  and  y,  will  have  the  form: 

Axy  +  Bx(l  -  y)  +  C(l  -  x)y  +  D(l  -  x)(l  -  y) 
And  for  any  function,  f(x,  y},  the  coefficients  are  determined  by  the  law: 

f(x,y)  =/(!,  l).sy+/(l,0).*(l  -  30 +/(0,  !)•(!  -  x)y 

+/(0,0)-(1  -*)(!- y) 

100  See  Math.  An.  of  Logic,  pp.  60-69;  L.  of  T.,  pp.  71-79;  "The  Calculus  of  Logic," 
Cambridge  and  Dublin  Math.  Jour.,  in,  188-89.  That  this  same  method  of  development 
should  belong  both  to  Boole's  algebra  and  to  the  remodeled  algebra  of  logic,  in  which  + 
is  not  completely  disjunctive,  is  at  first  surprising.  But  a  completely  developed  function, 
in  either  algebra,  is  always  a  sum  of  terms  any  two  of  which  have  nothing  in  common. 
This  accounts  for  the  identity  of  form  where  there  is  a  real  and  important  difference  in  the 
meaning  of  the  symbols. 


The  Development  of  Symbolic  Logic  59 

Thus  if  f(x,  y}  =  ax  +  2by, 

/(I,  1)  =  o-l  +  26-l  =  a  +  26 

/(I,  0)  =  a-1  +  26-0  =  a 

/(O,  1)  =  a-0  +  26-1  =  26 

/(0,0)  =  a-0  +  26-0  =  0 
Hence  f(x,  y)  =  (a  +  2b)xy  +  ax(l  —  y]  +  26(1  —  x)y 

An  exactly  similar  law  governs  the  expansion  and  the  determination  of 
coefficients,  for  functions  of  any  number  of  variables.  In  the  words  of 
Boole:101 

"The  general  rule  of  development  will  .  .  .  consist  of  two  parts,  the 
first  of  which  will  relate  to  the  formation  of  the  constituents  of  the  expansion, 
the  second  to  the  determination  of  their  respective  coefficients.  It  is  as 
follows : 

"  1st.  To  expand  any  function  of  the  symbols  x,  y,  2— Form  a  series 
of  constituents  in  the  following  manner:  Let  the  first  constituent  be  the 
product  of  the  symbols:  change  in  this  product  any  symbol  z  into  1  —  z, 
for  the  second  constituent.  Then  in  both  these  change  any  other  symbol 
y  into  1  —  y,  for  two  more  constituents.  Then  in  the  four  constituents 
thus  obtained  change  any  other  symbol  x  into  1  —  x,  for  four  new  constit- 
uents, and  so  on  until  the  number  of  possible  changes  has  been  exhausted. 

"2ndly.  To  find  the  coefficient  of  any  constituent — If  that  constituent 
involves  £  as  a  factor,  change  in  the  original  function  x  into  1;  but  if  it 
involves  1  —  x  as  a  factor,  change  in  the  original  function  x  into  0.  Apply 
the  same  rule  with  reference  to  the  symbols  y,  z,  etc. :  the  final  calculated 
value  of  the  function  thus  transformed  will  be  the  coefficient  sought." 

Two  further  properties  of  developed  functions,  which  are  useful  in 
solutions  and  interpretations,  are:  (1)  The  product  of  any  two  constituents 
is  0.  If  one  constituent  be,  for  example,  xyz,  any  other  constituent  will 
have  as  a  factor  one  or  more  of  the  negatives,  1  —  x,  I  —  y,  1  —  z. 
Thus  the  product  of  the  two  will  have  a  factor  of  the  form  x(l  —  x).  And 
where  x  is  a  "logical  symbol ",  susceptible  only  of  the  values  0  and  1,  x(l  —  x) 
is  always  0.  And  (2)  if  each  constituent  of  any  expansion  have  the  coef- 
ficient 1,  the  sum  of  all  the  constituents  is  1. 

All  information  which  it  may  be  desired  to  obtain  from  a  given  set  of 
premises,  represented  by  equations,  will  be  got  either  (1)  by  a  solution,  to 
determine  the  equivalent,  in  other  terms,  of  some  "logical  symbol"  x,  or 

101 L.  of  T.,  pp.  75-76. 


60  A  Survey  of  Symbolic  Logic 

(2)  by  an  elimination,  to  discover  what  statements  (equations),  which  are 
independent  of  some  term  x,  are  warranted  by  given  equations  which  in- 
volve x,  or  (3)  by  a  combination  of  these  two,  to  determine  the  equivalent 
of  x  in  terms  of  t,  u,  v,  from  equations  which  involve  x,  t,  u,  v,  and  some 
other  "logical"  symbol  or  symbols  which  must  be  eliminated  in  the  desired 
result.  "  Formal "  reasoning  is  accomplished  by  the  elimination  of  "middle" 
terms. 

The  student  of  symbolic  logic  in  its  current  form  knows  that  any  set 
of  equations  may  be  combined  into  a  single  equation,  that  any  equation 
involving  a  term  x  may  be  given  the  form  Ax  +  B(l  —  x)  =  0,  and  that 
the  result  of  eliminating  x  from  such  an  equation  is  AB  =  0.  Also,  the 
solution  of  any  such  equation,  provided  the  condition  AB  =  0  be  satisfied, 
will  be  x  =  B  +  v(l  —  A),  where  v  is  undetermined.  Boole's  methods 
achieve  these  same  results,  but  the  presence  of  numerical  coefficients  other 
than  0  and  1,  as  well  as  the  inverse  operations  of  subtraction  and  division, 
necessarily  complicates  his  procedure.  And  he  does  not  present  the  matter 
of  solutions  in  the  form  in  which  we  should  expect  to  find  it  but  in  a  more 
complicated  fashion  which  nevertheless  gives  equivalent  results.  We  have 
now  to  trace  the  procedures  of  interpretation,  reduction,  etc.  by  which 
Boole  obviates  the  difficulties  of  his  algebra  which  have  been  mentioned. 

The  simplest  form  of  equation  is  that  in  which  a  developed  function, 
of  any  number  of  variables,  is  equated  to  0,  as: 

Ax  +  B(l  -  x)  =  0,         or, 
Axy  +  Bx(l  -  y)  +  C(l  -  x)y+D(l  -  *)(!  -  y)  =  0,         etc. 

It  is  an  important  property  of  such  equations  that,  since  the  product  of 
any  two  constituents  in  a  developed  function  is  0,  any  such  equation  gives 
any  one  of  its  constituents,  whose  coefficient  does  not  vanish  in  the  develop- 
ment, =  0.  For  example,  if  we  multiply  the  second  of  the  equations  given 
by  xy,  all  constituents  after  the  first  will  vanish,  giving  Axy  =  0.  Whence 
we  shall  have  xy  =  0. 

Any  equation  in  which  a  developed  function  is  equated  to  1  may  be 
reduced  to  the  form  in  which  one  member  is  0  by  the  law;  If  V  =  1, 
1-7-0. 

The  more  general  form  of  equation  is  that  in  which  some  "logical 
symbol",  w,  is  equated  to  some  function  of  such  symbols.  For  example, 
suppose  x  =  yz,  and  it  be  desired  to  interpret  z  as  a  function  of  x  and  y. 
x  =  yz  gives  z  =  x/y;  but  this  form  is  not  interpretable.  We  shall,  then, 


The  Development  of  Symbolic  Logic  61 

develop  x/y  by  the  law 

f(x,  y}  =  /(I,  1)  -*y  +/(!,  0)  •*(!  -  </)  +/(0,  !)•(!-  *)y 

+  /(0,0).(1  -x)(l  -y} 
By  this  law : 

x 
If  2  =  -  ,  then 

y 

3  =  xy  +  -x(l  -  2/)  +  0(l  -  aOy+  -  (1  -  or)(l  -  z/) 

These  fractional  coefficients  represent  the  sole  necessary  difference  of  Boole's 
methods  from  those  at  present  familiar — a  difference  due  to  the  presence 
of  division  in  his  system.  Because  any  function  can  always  be  de- 
veloped, and  the  difference  between  any  two  developed  functions,  of  the 
same  variables,  is  always  confined  to  the  coefficients.  If,  then,  we  can 
interpret  and  successfully  deal  with  such  fractional  coefficients,  one  of  the 
difficulties  of  Boole's  system  is  removed. 

The  fraction  0/0  is  indeterminate,  and  this  suggests  that  a  proper  inter- 
pretation of  the  coefficient  0/0  would  be  to  regard  it  as  indicating  an  unde- 
termined portion  of  the  class  whose  coefficient  it  is.  This  interpretation 
may  be  corroborated  by  considering  the  symbolic  interpretation  of  "All 
x  is  y",  which  is  x(l  —  y)  =  0. 

If  x(l  —  y}  =  0,  then  x  —  xy  —  0  and  x  =  xy. 
Whence  y  =  x/x. 

Developing  x/x,  we  have  y  =  x  +  -  (1  —  x). 

If  "All  x  is  y",  the  class  y  is  made  up  of  the  class  x  plus  an  undetermined 
portion  of  the  class  not-x.  Whence  0/0  is  equivalent  to  an  arbitrary 
parameter  v,  which  should  be  interpreted  as  "an  undetermined  portion  of" 
or  as  "All,  some,  or  none  of". 

The  coefficient  1/0  belongs  to  the  general  class  of  symbols  which  do  not 
obey  the  "index  law",  x2  =  x,  or  its  equivalent,  the  "law  of  duality", 
x(l  —  x)  =  0.  At  least  Boole  says  it  belongs  to  this  class,  though  the 
numerical  properties  of  1/0  would,  in  fact,  depend  upon  laws  which  do  not 
belong  to  Boole's  system.  But  in  any  case,  1/0  belongs  with  the  class  of 
such  coefficients  so  far  as  its  logical  interpretation  goes.  Any  constituent  of  a 
developed  function  which  does  not  satisfy  the  index  law  must  be  separately 
equated  to  0.  Suppose  that  in  any  equation 

w  =  At  +  P 


62  A  Survey  of  Symbolic  Logic 

w  be  a  "logical  symbol",  and  t  be  a  constituent  of  a  developed  function 
whose  coefficient  A  does  not  satisfy  the  index  law,  A2  =  A.  And  let  P 
be  the  sum  of  the  remaining  constituents  whose  coefficients  do  satisfy  this 
law.  Then 

wz  =  W)        t2  =  t,        and        P2  =  P 

Since  the  product  of  any  two  constituents  of  a  development  is  0, 

w2  =  (At+P)z  =  AW  +  P2 
Hence  w  =  AH  +  P 

Subtracting  this  from  the  original  equation, 

(A  -  A*)t  =  0  =  A(l  -  A)t 
Hence  since  A(l  -  A)  4=  0,        t  =  0 

Hence  any  equation  of  the  form 


is  equivalent  to  the  two  equations 

w  =  P  +  vR        and        S  =  0 

which  together  represent  its  complete  solution. 

It  will  be  noted  that  a  fraction,  in  Boole's  algebra,  is  always  an  am- 
biguous function.  Hence  the  division  operation  must  never  be  performed: 
the  value  of  a  fraction  is  to  be  determined  by  the  law  of  development  only, 
except  for  the  numerical  coefficients,  which  are  elsewhere  discussed.  We 
have  already  remarked  that  ax  =  bx  does  not  give  a  =  b,  because  x  may 
have  the  value  0.  But  we  may  transform  ax  =  bx  into  a  =  bx/x  and 
determine  this  fraction  by  the  law 

f(b,x)  =/(!,  l).te+/(l,0)-6(l  -*)+/(0,  !)•(!  -b)x 

+/(0,0).(1  -&)(!  -*) 
We  shall  then  have 


and  this  is  not,  in  general,  equivalent  to  b.     Replacing  0/0  by  indeterminate 
coefficients,  v  and  w,  this  gives  us, 
If  ax  =  bx,  then 

a  =  bx  +  vb(l  —  x)  +  wfl  —  6)(1  —  x} 


The  Development  of  Symbolic  Logic  63 

And  this  result  is  always  valid.     Suppose,  for  example,  the  logical  equation 
rational  men  =  married  men 

and  suppose  we  wish  to  discover  who  are  the  rational  beings.     Our  equation 
will  not  give  us 

rational  =  married 
but  instead 

rational  =  married  men  +  v  -  married  non-men  +  w  •  non-married  non-men 
That  is,  our  hypothesis  is  satisfied  if  the  class  "rational  beings"  consist  of 
the  married  men  together  with  any  portion  (which  may  be  null)  of  the 
class  "married  women"  and  a  similarly  undetermined  portion  of  the  class 
"unmarried  women". 

If  we  consider  Boole's  system  as  an  algebra  of  0  and  1,  and  the  fact  that 
for  any  fraction,  xjy, 

x  1  0 

-  =  xy  +  -x(l  -  y)  +  ~(1  -aO(l  -  y) 

we  shall  find,  by  investigating  the  cases 

(1)  x  =  1  and  y  =  1;         (2)  x  =  1,  y  =  0;         (3)  x  =  0,  y  =  1; 

and         (4)  x  =  0,  y  =  0, 
that  it  requires  these  three  possible  cases: 


Or,  to  speak  more  accurately,  it  requires  that  0/0  be  an  ambiguous  function 
susceptible  of  the  values  0  and  1. 

Since  there  are,  in  general,  only  four  possible  coefficients,  1,  0,  0/0,  and 
such  as  do  not  obey  the  index  law,  of  which  1/0  is  a  special  case,  this  means 
that  any  equation  can  be  interpreted,  and  the  difficulty  due  to  the  presence 
of  an  uninterpretable  division  operation  in  the  system  has  disappeared. 
And  any  equation  can  be  solved  for  any  "logical  symbol"  x,  by  trans- 
ferring all  other  terms  to  the  opposite  side  of  the  equation,  by  subtraction 
or  division  or  both,  and  developing  that  side  of  the  equation. 

Any  equation  may  be  put  in  the  form  in  which  one  member  is  0  by 


64  A  Survey  of  Symbolic  Logic 

bringing  all  the  terms  to  one  side.  When  this  is  done,  and  the  equation 
fully  expanded,  all  the  coefficients  which  do  not  vanish  may  be  changed  to 
unity,  except  such  as  already  have  that  value.  Boole  calls  this  a  "rule  of 
interpretation".102  Its  validity  follows  from  two  considerations:  (1)  Any 
constituent  of  an  equation  with  one  member  0,  whose  coefficient  does  not 
vanish  in  development,  may  be  separately  equated  to  0;  (2)  the  sum  of 
the  constituents  thus  separately  equated  to  0  will  be  an  equation  with  one 
member  0  in  which  each  coefficient  will  be  unity. 

Negative  coefficients  may  be  eliminated  by  squaring  both  sides  of  any 
equation  in  which  they  appear.  The  "logical  symbols"  in  any  function 
are  not  altered  by  squaring,  and  any  expression  of  the  form  (1  —  x),  where 
#  is  a  "logical  symbol",  is  not  altered,  since  it  can  have  only  the  values 
0  and  1  .  Hence  no  constituent  is  altered,  except  that  its  coefficient  may  be 
altered.  And  any  negative  coefficient  will  be  made  positive.  No  new 
terms  will  be  introduced  by  squaring,  since  the  product  of  any  two  terms 
of  a  developed  function  is  always  null.  Hence  the  only  change  effected 
by  squaring  any  developed  function  is  the  alteration  of  any  negative  coef- 
ficients into  positive.  Their  actual  numerical  value  is  of  no  consequence, 
because  coefficients  other  than  1  can  be  dealt  with  by  the  method  described 
above. 

For  reducing  any  two  or  more  equations  to  a  single  equation,  Boole 
first  proposed  the  "method  of  indeterminate  multipliers",103  by  which 
each  equation,  after  the  first,  is  multiplied  by  an  arbitrary  constant  and  the 
equations  then  added.  But  these  indeterminate  multipliers  complicate  the 
process  of  elimination,  and  the  method  is,  as  he  afterward  recognized,  an 
inferior  one.  More  simply,  such  equations  may  be  reduced,  by  the  methods 
just  described,  to  the  form  in  which  one  member  is  0,  and  each  coefficient 
is  1.  They  may  then  be  simply  added;  the  resulting  equation  will  combine 
the  logical  significance  of  the  equations  added. 

Any  "logical  symbol"  which  is  not  wanted  in  an  equation  may  be 
eliminated  by  the  method  which  is  familiar  to  all  students  of  symbolic 
logic.  To  eliminate  x,  the  equation  is  reduced  to  the  form 

-  x}  =  0 


The  result  of  elimination  will  be104 

AB  =  0 

102  L.  of  T.,  p.  90. 

103  See  Math.  An.  of  Logic,  pp.  78-81;  L.  of  T.,  pp.  115-120. 

104  See  L.  of  T.,  p.  101.     We  do  not  pause  upon  this  or  other  matters  which  will  be 
entirely  clear  to  those  who  understand  current  theory. 


The  Development  of  Symbolic  Logic  65 

By  these  methods,  the  difference  between  Boole's  algebra  and  the  classic 
algebra  of  logic  which  grew  out  of  it  is  reduced  to  a  minimum.  Any  logical 
results  obtainable  by  the  use  of  the  classic  algebra  may  also  be  got  by 
Boole's  procedures.  The  difference  is  solely  one  of  ease  and  mathematical 
neatness  in  the  method.  Two  important  laws  of  the  classic  algebra  which 
do  not  appear  among  Boole's  principles  are: 

(1)  x  +  x  =  x,        and         (2)  x  =  x  +  xy 

These  seem  to  be  inconsistent  with  the  Boolean  meaning  of  +  ;  the  first  of 
them  does  not  hold  for  x  =  1;  the  second  does  not  hold  for  a-  =  1,  y  =  1. 
But  although  they  do  not  belong  to  Boole's  system  as  an  abstract  algebra, 
the  methods  of  reduction  which  have  been  discussed  will  always  give  x  in 
place  of  x  +  x  or  of  x  +  xy,  in  any  equation  in  which  these  appear.  The 
expansion  of  x  +  x  gives  2x;  the  expansion  of  x  +  xy  gives  2xy  +  x(l  —  y). 
By  the  method  for  dealing  with  coefficients  other  than  unity,  2x  may  be 
replaced  in  the  equation  by  x,  and  2xy  +  x(l  —  y}  by  xy  +  x(l  —  y},  which 
is  equal  to  x. 

The  methods  of  applying  the  algebra  to  the  relations  of  logical  classes 
should  now  be  sufficiently  clear.  The  application  to  propositions  is  made 
by  the  familiar  device  of  correlating  the  "logical  symbol",  x,  with  the 
times  when  some  proposition,  X.  is  true,  xy  will  represent  the  times  wrhen 
X  and  Y  are  both  true;  x(l  —  y},  the  times  when  X  is  true  and  Y  is  false, 
and  so  on.  Congruent  with  the  meaning  of  +  ,  x  +  y  will  represent  the 
times  when  either  X  or  Y  (but  not  both)  is  true.  In  ord°r  to  symbolize 
the  times  when  X  or  Y  or  both  are  true,  we  must  write  x  +  (1  —  x)y,  or 
xy  +  x(l  —  y*)  +  (1  —  x)y.  1,  the  "universe",  will  represent  "all  times"  or 
"always";  and  0  will  be  "no  time"  or  "never",  x  =  1  will  represent 
"X  is  always  true";  x  =  0  or  (1  —  x)  =  1,  "X  is  never  true,  is  always 
false". 

Just  as  there  is,  with  Boole,  no  symbol  for  the  inclusion  relation  of 
classes,  so  there  is  no  symbol  for  the  implication  relation  of  propositions. 
For  classes,  "All  X  is  7"  or  " X  is  contained  in  Y"  becomes  x  =  vy.  Cor- 
respondingly, "All  times  when  X  is  true  are  times  when  Y  is  true"  or  "If 
A"  then  Y"  or  "X  implies  Y"  is  x  =  vy.  x  =  y  will  mean,  "The  times 
when  X  is  true  and  the  times  when  Y  is  true  are  the  same"  or  " X  implies 
Yand  Y  implies  X". 

The  entire  procedure  for  "secondary  propositions"  is  summarized  as 
follows:105 

i"5  L.  of  T.,  p.  178. 


66  A  Survey  of  Symbolic  Logic 

"Rule. — Express  symbolically  the  given  propositions.  .  .  . 
"Eliminate  separately  from  each  equation  in  which  it  is  found  the 

indefinite  symbol  «. 

"Eliminate  the  remaining  symbols  which  it  is  desired  to  banish  from 

the  final  solution :  always  before  elimination,  reducing  to  a  single  equation 

those  equations  in  which  the  symbol  or  symbols  to  be  eliminated  are  found. 

Collect  the  resulting  equations  into  a  single  equation  [one  member  of  which 

is  0],  F  =  0. 

"Then  proceed  according  to  the  particular  form  in  which  it  is  desired 

to  express  the  final  relation,  as 

1st.  If  in  the  form  of  a  denial,  or  system  of  denials,  develop  the 
function  V,  and  equate  to  0  all  those  constituents  whose  coefficients  do 
not  vanish. 

2ndly.  If  in  the  form  of  a  disjunctive  proposition,  equate  to  1  the 
sum  of  those  constituents  whose  coefficients  vanish. 

3rdly.  If  in  the  form  of  a  conditional  proposition  having  a  simple 
element,  as  x  or  1  —  x,  for  its  antecedent,  determine  the  algebraic 
expression  of  that  element,  and  develop  that  expression. 

4thly.  If  in  the  form  of  a  conditional  proposition  having  a  com- 
pound expression,  as  xy,  xy+  (I  —  .r)(l  --  y),  etc.,  for  its  antecedent, 
equate  that  expression  to  a  new  symbol  t,  and  determine  t  as  a  developed 
function  of  the  symbols  which  are  to  appear  in  the  consequent.  .  .  . 

5thly.  ...  If  it  only  be  desired  to  ascertain  whether  a  particular 
elementary  proposition  x  is  true  or  false,  we  must  eliminate  all  the 
symbols  but  x;  then  the  equation  x  =  1  will  indicate  that  the  proposi- 
tion is  true,  x  =  0  that  it  is  false,  0  =  0  that  the  premises  are  insuf- 
ficient to  determine  whether  it  is  true  or  false." 

It  is  a  curious  fact  that  the  one  obvious  law  of  an  algebra  of  0  and  1 
which  Boole  does  not  assume  is  exactly  the  law  which  would  have  limited 
the  logical  interpretation  of  his  algebra  to  propositions.  The  law 

If  x  =}=  1>  x  =  0  and  if  x  =(=  0,  x  =  1 

is  exactly  the  principle  which  his  successors  added  to  his  system  when  it 
is  to  be  considered  as  a  calculus  of  propositions.  This  principle  would  have 
made  his  system  completely  inapplicable  to  logical  classes. 

For  propositions,  this  principle  means,  "  If  x  is  not  true,  then  x  is  false, 
and  if  x  is  not  false,  it  is  true".  But  careful  attention  to  Boole's  interpre- 
tation for  "propositions"  shows  that  in  his  system  x  =  0  should  be  inter- 


The  Development  of  Symbolic  Logic  67 

preted  "x  is  false  at  all  times  (or  in  all  cases)",  and  x  =  1  should  be  in- 
terpreted "x  is  true  at  all  times".  This  reveals  that  fact  that  what  Boole 
calls  "  propositions  "  are  what  we  should  now  call  "  prepositional  functions  ", 
that  is,  statements  which  may  be  true  under  some  circumstances  and  false 
under  others.  The  limitation  put  upon  what  we  now  call  "  propositions  "- 
namely  that  they  must  be  absolutely  determinate,  and  hence  simply  true 
or  false — does  not  belong  to  Boole's  system.  And  his  treatment  of  "prepo- 
sitional symbols"  in  the  application  of  the  algebra  to  probability  theory 
gives  them  the  character  of  " prepositional  functions"  rather  than  of  our 
absolutely  determinate  propositions. 

The  last  one  hundred  and  seventy-five  pages  of  the  Laws  of  Thought 
are  devoted  to  an  application  of  the  algebra  to  the  solution  of  problems  in 
probabilities.106  This  application  amounts  to  the  invention  of  a  new 
method — a  method  whereby  any  logical  analysis  involved  in  the  problem 
is  performed  as  automatically  as  the  purely  mathematical  operations. 
We  can  make  this  provisionally  clear  by  a  single  illustration : 

All  the  objects  belonging  to  a  certain  collection  are  classified  in  three 
ways — as  ^4's  or  not,  as  B's  or  not,  and  as  C's  or  not.  It  is  then  found 
that  (1)  a  fraction  m/n  of  the  ^4's  are  also  B's  and  (2)  the  C"s  consist  of  the 
^4's  which  are  not  B's  together  with  the  B's  which  are  not  A's. 

Required:  the  probability  that  if  one  of  the  A's  be  taken  at  random, 
it  will  also  be  a  C. 

By  premise  (2) 

C  =  A(l  -  £)  +  (!  -  A)B 

Since  A,  B,  and  C  are  "logical  symbols",  A*  =  A  and  A(l  —  A}  =  0. 
Hence,  AC  =  A\l  -  5) +  .4(1  -  A)B  =  A(\  -  B). 

The  A's  which  are  also  C"s  are  identical  with  the  ^4's  which  are  not  B's. 
Thus  the  probability  that  a  given  A  is  also  a  C  is  exactly  the  probability 
that  it  is  not  a  B;  or  by  premise  (1),  1  —  m/n,  which  is  the  required  solution. 

In  any  problem  concerning  probabilities,  there  are  usually  two  sorts  of 
difficulties,  the  purely  mathematical  ones,  and  those  involved  in  the  logical 
analysis  of  the  situation  upon  which  the  probability  in  question  depends. 
The  methods  of  Boole's  algebra  provide  a  means  for  expressing  the  relations 
of  classes,  or  events,  given  in  the  data,  and  then  transforming  these  logical 

106  Chap.  16  ff.  See  also  the  Keith  Prize  essay  "On  the  Application  of  the  Theory  of 
Probabilities  to  the  Question  of  the  Combination  of  Testimonies  or  Judgments",  Trans. 
Roy.  Soc.  Edinburgh,  xxi,  597  ff.  Also  a  series  of  articles  in  Phil.  Mag.,  1851-54  (see 
Bibl).  An  article  on  the  related  topic  "Of  Propositions  Numerically  Definite"  appeared 
posthumously;  Camb.  Phil.  Trans.,  xr,  396-411. 


68  A  Survey  of  Symbolic  Logic 

equations  so  as  to  express  the  class  which  the  quaesitum  concerns  as  a  func- 
tion of  the  other  classes  involved.  It  thus  affords  a  method  for  untangling 
the  problem — often  the  most  difficult  part  of  the  solution. 

The  parallelism  between  the  logical  relations  of  classes  as  expressed  in 
Boole's  algebra  and  the  corresponding  probabilities,  numerically  expressed, 
is  striking.  Suppose  x  represent  the  class  of  cases  (in  a  given  total)  in  which 
the  event  X  occurs — or  those  which  "are  favorable  to"  the  occurrence  of 
X.107  And  let  p  be  the  probability,  numerically  expressed,  of  the  event  X. 
The  total  class  of  cases  will  constitute  the  logical  "universe",  or  1;  the 
null  class  will  be  0.  Thus,  if  x  =  1 — if  all  the  cases  are  favorable  to  X— 
then  p  =  1 — the  probability  of  X  is  "certainty".  If  x  =  0,  then  p  =  0. 
Further,  the  class  of  cases  in  which  X  does  not  occur,  will  be  expressed  by 
1  —  x;  the  probability  that  X  will  not  occur  is  the  numerical  1  —  p.  Also, 
x+  (1  —  x}  =  1  and  p+  (1  —  p*)  =  1. 

This  parallelism  extends  likewise  to  the  combinations  of  two  or  more 
events.  If  x  represent  the  class  of  cases  in  which  X  occurs,  and  y  the  class 
of  cases  in  which  Y  occurs,  then  xy  will  be  the  class  of  cases  in  which  X 
and  7  both  occur;  #(1  —  y},  the  cases  in  which  X  occurs  without  Y; 
(1  —  x)y,  the  cases  in  which  Y  occurs  without  X;  (1  —  z)(l  —  y),  the 
cases  in  which  neither  occurs;  x(l  —  y}  +  y(l  —  x),  the  cases  in  which 
X  or  7  occurs  but  not  both,  and  so  on.  Suppose  that  X  and  Y  are  "  simple  " 
and  "independent"  events,  and  let  p  be  the  probability  of  X,  q  the  prob- 
ability of  y.  Then  we  have: 

Combination  of  events  Corresponding  probabilities 

expressed  in  Boole's  algebra  numerically  expressed 

xy  pq 

x(l  -  y}  p(l  -  q) 

(1  -  x}y  (1  -  q)p 

(1  -  .r)(l  -  y)  (I-  p)(l  -  <?) 

x(l  -  */)  +  (!  -  x}y  p(l  -  q)  +  (1  -  p)q 

Etc.  etc. 

In  fact,  this  parallelism  is  complete,  and  the  following  rule  can  be 
formulated:108 

107  Boole  prefers  to  consider  x  as  representing  the  times  when  a  certain  proposition, 
asserting  an  occurrence,  will  be  true.     But  this  interpretation  comes  to  exactly  the  same 
thing. 

108  L.  of  T.,  p.  258. 


The  Development  of  Symbolic  Logic  69 

"If  p,  q,  r,  .  .  .  are  the  respective  probabilities  of  unconditioned  simple 
events,  x,  y,  z,  .  .  .  ,  the  probability  of  any  compound  event  F  will  be  [F], 
this  function  [F]  being  formed  by  changing,  in  the  function  F,  the  symbols 
x,  y,  z,  .  .  .  into  p,  q,  r,  .  .  .  . 

"According  to  the  well-known  law  of  Pascal,  the  probability  that  if 
the  event  F  occur,  the  event  V  will  occur  with  it,  is  expressed  by  a  fraction 
whose  numerator  is  the  probability  of  the  joint  occurrence  of  F  and  V, 
and  whose  denominator  is  the  probability  of  the  occurrence  of  F.  We  can 
then  extend  the  rule  just  given  to  such  cases : 

"The  probability  that  if  the  event  F  occur,  any  other  event  V  wTill 

[FF'j 
also  occur,  will  be  ,  where  [F  F']  denotes  the  result  obtained  by 

multiplying  together  the  logical  functions  F  and  V ,  and  changing  in  the 
result  x,  y,  z,  .  .  .  into  p,  q,  r,  .  .  .  ." 

The  inverse  problem  of  finding  the  absolute  probability  of  an  event 
when  its  probability  upon  a  given  condition  is  known  can  also  be  solved. 

Given:  The  probabilities  of  simple  events  x,  y,  z,  .  .  .  are  respectively 
p,  q,  r,  .  .  .  when  a  certain  condition  F  is  satisfied. 

To  determine :  the  absolute  probabilities  I,  m,  n,  .  .  .  of  x,  y,  z,  .  .  .  . 

By  the  rule  just  given, 

[xV\  '  [yV]  [zV] 

=    Tt  r=    n  =    T  PTP 

[F]  [FJ  [F] 

And  the  number  of  such  equations  will  be  equal  to  the  number  of  unknowns, 
I,  m,  n,  .  .  .  to  be  determined.109  The  determination  of  any  logical  expres- 
sion of  the  form  xV  is  peculiarly  simple  since  the  product  of  x  into  any 
developed  function  F  is  the  sum  of  those  constituents  of  F  which  contain  x 
as  a  factor.  For  example : 

if    F  =  xyz  +  x(l  —  y)z  +  (1  —  x)y(l  —  z)  +  (1  —  x)  (1  —  y)z, 
xV  =  xyz  +  x(l  —  y)z 
yV  =  xyz  +  (1  -  x)y(l  -  z} 
zV  =  xyz  +  x(l  —  y)z  +  (1  —  z)(l  —  y}z 
Thus  any  equation  of  the  form 

[xV] 


109  On  certain  difficulties  in  this  connection,  and  their  solution,  see  Cayley,  "On  a 
Question  in  the  Theory  of  Probability"  (with  discussion  by  Boole),  Phil.  Mag.,  Ser.  IV, 
xxm  (1862),  352-65,  and  Boole,  "On  a  General  Method  in  the  Theory  of  Probabilities", 
ibid.,  xxv  (1863),  313-17. 


70  A  Survey  of  Symbolic  Logic 

is  readily  determined  as  a  numerical  equation.     Boole  gives  the  following 
example  in  illustration:  ll° 

"Suppose  that  in  the  drawings  of  balls  from  an  urn,  attention  had  only 
been  paid  to  those  cases  in  which  the  balls  drawn  were  either  of  a  particular 
color,  'white,'  or  of  a  particular  composition,  'marble,'  or  were  marked  by 
both  of  these  characters,  no  record  having  been  kept  of  those  cases  in  which 
a  ball  which  was  neither  white  nor  of  marble  had  been  drawn.  Let  it  then 
have  been  found,  that  whenever  the  supposed  condition  was  satisfied,  there 
was  a  probability  p  that  a  white  ball  would  be  drawn,  and  a  probability  q 
that  a  marble  ball  would  be  drawn:  and  from  these  data  alone  let  it  be 
required  to  find  the  probability  m  that  in  the  next  drawing,  without  refer- 
ence at  all  to  the  condition  above  mentioned,  a  white  ball  will  be  drawn; 
&lso  a  probability  n  that  a  marble  ball  will  be  drawn. 

"Here  if  x  represent  the  drawing  of  a  white  ball,  y  that  of  a  marble 
ball,  the  condition  V  will  be  represented  by  the  logical  function 

xy  +  x(l  -  y)+(l  -  x)y 
Hence  we  have 

xV  =  xy  +  x(l  —  y)  =  x 

yV  =  xy+(l  -  x)y  =  y 
Whence 

[xV]  =  m,         [yV]  =  n 

and  the  final  equations  of  the  problem  are 

m 
mn  +  ra(l  —  n)  +  (1  —  m)n 

n 

mn  +  m(l  —  n)  +  (1  —  m)n 
from  which  we  find 

p+  q  —  1  p  +  q  —  I 

m  =  -  ,         n  =  - 

q  p 

...  To  meet  a  possible  objection,  I  here  remark  that  the  above  reasoning 
does  not  require  that  the  drawings  of  a  white  and  a  marble  ball  should  be 
independent,  in  virtue  of  the  physical  constitution  of  the  balls. 

"In  general,  the  probabilities  of  any  system  of  independent  events 
being  given,  the  probability  of  any  event  X  may  be  determined  by  finding  a 
logical  equation  of  the  form 

x  =  A+OB  +  ^c+  ID 

110  L.  of  T.,  p.  262.     I  have  slightly  altered  the  illustration  by  a  change  of  letters. 


The  Development  of  Symbolic  Logic  71 

where  A,  B,  C,  and  D  are  functions  of  the  symbols  of  the  other  events. 
As  has  already  been  shown,  this  is  the  general  type  of  the  logical  equation, 
and  its  interpretation  is  given  by 

x  =  A  +  vC,        where  v  is  arbitrary  and 
D  =  0 

By  the  properties  of  constituents,  we  have  also  the  equation, 

A  +  B+C  +  D  =  1 
and ,  since  D  =  0, 

A+B  +  C  =  1 

A  +  B  +  C  thus  gives  the  '  universe '  of  the  events  in  question,  and  the  prob- 
abilities given  in  the  data  are  to  be  interpreted  as  conditioned  by  A  +  B  +  C 
=  1,  since  D  =  0  is  the  condition  of  the  solution  x  =  A  +  vC.  If  the  given 
probability  of  some  event  S  is  p,  of  T  is  q,  etc.,  then  the  supposed  'absolute' 
probabilities  of  S,  T,  etc.,  may  be  determined  by  the  method  which  has 
been  described.  Let  V  =  A  +  B  +  C,  then 

[»V]  [tV] 

[V]  "  [V]  " 

where  [sV],  [tV],  etc.  are  the  "absolute  probabilities"  sought.  These, 
being  determined,  may  be  substituted  in  the  equation 

\A  +  vC] 
Prob.  w  =  — 


which  will  furnish  the  required  solution. 

"The  term  vC  will  appear  only  in  cases  where  the  data  are  insufficient 
to  determine  the  probability  sought.  Where  it  does  appear,  the  limits  of 
this  probability  may  be  determined  by  giving  v  the  limiting  values,  0  and  1. 
Thus 

[A] 


Lower  limit  of  Prob.  w  = 
Upper  limit 


[V] 

[A  +  C] 


[V] 

With  the  detail  of  this  method,  and  with  the  theoretical  difficulties  of 
its  application  and  interpretation,  we  need  not  here  concern  ourselves. 
Suffice  it  to  say  that,  with  certain  modifications,  it  is  an  entirely  workable 
method  and  seems  to  possess  certain  marked  advantages  over  those  more 
generally  in  use.  It  is  a  matter  of  surprise  that  this  immediately  useful 
application  of  symbolic  logic  has  been  so  generally  overlooked. 


72  A  Survey  of  Symbolic  Logic 

VI.    JEVONS 

It  has  been  shown  that  Boole's  "calculus  of  logic"  is  not  so  much  a 
system  of  logic  as  an  algebra  of  the  numbers  0  and  1,  some  of  whose  ex- 
pressions are  capable  of  simple  interpretation  as  relations  of  logical  classes, 
or  propositions,  and  some  of  whose  transformations  represent  processes  of 
reasoning.  If  the  entire  algebra  can,  with  sufficient  ingenuity,  be  inter- 
preted as  a  system  of  logic,  still  Boole  himself  failed  to  recognize  this  fact, 
and  this  indicates  the  difficulty  and  unnaturalness  of  some  parts  of  this 
interpretation. 

Jevons111  pointed  a  way  to  the  simplification  of  Boole's  algebra,  dis- 
carding those  expressions  which  have  no  obvious  interpretation  in  logic, 
and  laying  down  a  procedure  which  is  just  as  general  and  is,  in  important 
respects,  superior.  In  his  first  book  on  this  subject,  Jevons  says: 112 

"So  long  as  Professor  Boole's  system  of  mathematical  logic  was  capable 
of  giving  results  beyond  the  power  of  any  other  system,  it  had  in  this  fact 
an  impregnable  stronghold.  Those  who  were  not  prepared  to  draw  the 
same  inferences  in  some  other  manner  could  not  quarrel  with  the  manner 
of  Professor  Boole.  But  if  it  be  true  that  the  system  of  the  foregoing 
chapters  is  of  equal  power  with  Professor  Boole's  system,  the  case  is  altered. 
There  are  now  twro  systems  of  notation,  giving  the  same  formal  results,  one 
of  which  gives  them  with  self-evident  force  and  meaning,  the  other  by  dark 
and  symbolic  processes.  The  burden  of  proof  is  shifted,  and  it  must  be 
for  the  author  or  supporters  of  the  dark  system  to  show  that  it  is  in  some 
way  superior  to  the  evident  system." 

He  sums  up  the  advantages  of  his  system,  compared  with  Boole's,  as 
follows : 113 

"  1 .  Every  process  is  of  self-evident  nature  and  force,  and  governed  by 
laws  as  simple  and  primary  as  those  of  Euclid's  axioms. 

"2.  The  process  is  infallible,  and  gives  us  no  uninterpretable  or  anom- 
alous results. 

"3.  The  inferences  may  be  drawn  with  far  less  labor  than  in  Professor 
Boole's  system,  which  generally  requires  a  separate  computation  and 
development  for  each  inference." 

111  William  Stanley  Jevons  (1835-1882),  B.A.,  M.A.  (London),  logician  and  economist; 
professor  of  logic  and  mental  and  moral  philosophy  and  Cobden  professor  of  political 
economy  in  Owens  College,  Manchester,  1866-76;    professor  of  political  economy,  Uni- 
versity College,  London,  1876-80. 

112  Pure  Logic,  or  the  Logic  of  Quality  apart  from  Quantity,  p.  75. 

113  Ibid.,  p.  74. 


The  Development  of  Symbolic  Logic  73 

The  third  of  these  observations  is  not  entirely  warranted.  Jevons 
unduly  restricts  the  operations  and  methods  of  Boole  in  such  wise  that 
his  own  procedure  is  often  cumbersome  and  tedious  where  Boole's  would 
be  expeditious.  Yet  the  honor  of  first  pointing  out  the  simplifications 
which  have  since  been  generally  adopted  in  the  algebra  of  logic  belongs  to 
Jevons. 

He  discards  Boole's  inverse  operations,  a  —  b  and  a/b,  and  he  interprets 
the  sum  of  a  and  b  as  "either  a  or  6,  where  a  and  b  are  not  necessarily 
exclusive  classes".  (We  shall  symbolize  this  relation  by  a  +  b:  Jevons  has 
A  +  B  or  A  •  I  -B.yu  Thus,  for  Jevons,  a  +  a  =  a,  whereas  for  Boole  a  +  a 
is  not  interpretable  as  any  relation  of  logical  classes,  and  if  it  be  taken  as 
an  expression  in  the  algebra  of  0  and  1,  it  obeys  the  usual  arithmetical  laws, 
so  that  a  +  a  =  2a.  As  has  been  indicated,  this  is  a  source  of  much  awk- 
ward procedure  in  Boole's  system.  The  law  a  +  a  =  a  eliminates  numerical 
coefficients,  other  than  0  and  1,  and  this  is  a  most  important  simplification. 

Jevons  supposes  that  the  fundamental  difference  between  himself  and 
Boole  is  that  Boole's  system,  being  mathematical,  is  a  calculus  of  things 
taken  in  their  logical  extension,  while  his  own  system,  being  "pure  logic", 
is  a  calculus  of  terms  in  intension.  It  is  true  that  mathematics  requires 
that  classes  be  taken  in  extension,  but  it  is  not  true  that  the  calculus  of 
logic  either  requires  or  derives  important  advantage  from  the  point  of  view 
of  intension.  Since  Jevons's  system  can  be  interpreted  in  extension  without 
the  slightest  difficulty,  we  shall  ignore  this  supposed  difference. 

The  fundamental  ideas  of  the  system  are  as  follows: 

(1)  a  b  denotes  that  which  is  both  a  and  b,  or  (in  intension)  the  sum 
of  the  meanings  of  the  two  terms  combined. 

(2)  a  +  b  denotes  that  which  is  either  a  or  b  or  both,  or  (in  intension)  a 
term  with  one  of  two  meanings.115 

(3)  a  =  b     a  is  identical  writh  6,  or  (in  intension)  a  means  the  same  as  b. 

(4)  -b     Not-6,  the  negative  of  b,  symbolized  in  Boole's  system  by 
1  -  6.116 

(5)  0     According  to  Jevons,  0  indicates  that  which  is  contradictory  or 
"excluded  from  thought".     He  prefers  it  to  appear  as  a  factor  rather  than 

114  A  +  B  in  Pure  Logic;  A'  \  '  B  in  the  other  papers.     (See  Bibl.) 

116  Jevons  would  add  "but  it  is  not  known  which".     (See  Pure  Logic,  p.  25.)     But 

this  is  hardly  correct;   it  makes  no  difference  if  it  is  known  which,  since  the  meaning  of 

a  +  b  does  not  depend  on  the  state  of  our  knowledge.     Perhaps  a  better  qualification  would 

be  "but  it  is  not  specified  which". 

116  Jevons  uses  capital  roman  letters  for  positive  terms  and  the  corresponding  small 

italics  for  their  negatives.     Following  De  Morgan,  he  calls  A  and  a  "contrary"  terms. 


74  A  Survey  of  Symbolic  Logic 

by  itself.117  The  meaning  given  is  a  proper  interpretation  of  the  symbol  in 
intension.  Its  meaning  in  extension  is  the  null-class  or  "nothing",  as  with 
Boole. 

•Jevons  does  not  use  any  symbol  for  the  "universe",  but  writes  out  the 
"logical  alphabet".  This  "logical  alphabet",  for  any  number  n  of  ele- 
ments, a,  b,  c,  .  .  .  ,  consists  of  the  2n  terms  which,  in  Boole's  system,  form 
the  constituents  of  the  expansion  of  1.  Thus,  for  two  elements,  a  and  6, 
the  "logical  alphabet"  consists  of  a  b,  a-b,  -ab,  and  -a-b.  For  three 
terms,  x,  y,  z,  it  consists  of  x  y  z,  x  y  -z,  x  -y  z,  x  -y  -z,  x  y  -z,  -x  -y  z,  and 
-x  -y  -z.  Jevons  usually  writes  these  in  a  column  instead  of  adding  them 
and  putting  the  sum  =  1 .  Thus  the  absence  of  1  from  his  system  is  simply 
a  whim  and  represents  no  real  difference  from  Boole's  procedure. 

The  fundamental  laws  of  the  system  of  Jevons  are  as  follows: 

(1)  If  a  =  b  and  b  —  c,  then  a  =  c. 

(2)  a  b  =  b  a. 

(3)  a  a  =  a. 

(4)  a  -a  =  0. 

(5)  a  +  b  =  b  +  a. 

(6)  a  +  a  =  a. 

(7)  a  +  0  =  a.     This  law  is  made  use  of  but  is  not  stated. 

(8)  a(b  +  c)  =  ab+  ac  and  (a  +  b)(c  +  d)  =  ac  +  ad+bc  +  bd. 

(9)  a  +  a  b  =  a.     This  law,  since  called  the  "law  of  absorption",  allows 
a  direct  simplification  which  is  not  possible  in  Boole.     Its  analogue  for 
multiplication 

a(a+  6)  =  a 

follows  from  (8),  (3),  and  (9).  The  law  of  absorption  extends  to  any 
number  of  terms,  so  that  we  have  also 

a  +  a b  +  ac  +  a b d  + .  .  .  =  a 

(10)  a  =  a(b  +  -b)(c  +  -c)  ....     This  is  the  rule  for  the  expansion  of 
any  term,  a,  with  reference  to  any  other  terms,  b,  c,  etc.     For  three  terms 
it  gives  us 

a  =  a(b  +•  -b)  (c  +  -c)  =  a  b  c  +  a  b  -c  +  a  -b  c  +  a  -b  -c 

This  expansion  is  identical  with  that  which  appears  in  Boole's  system,  except 
for  the  different  meaning  of  +  .  But  the  product  of  any  two  terms  of  such 
an  expansion  will  always  have  a  factor  of  the  form  a  -a,  and  hence,  by  (4), 
will  be  null.  Thus  the  terms  of  any  expansion  will  always  represent  classes 
117  See  Pure  Logic,  pp.  31-33. 


The  Development  of  Symbolic  Logic  75 

which  are  mutually  exclusive.  This  accounts  for  the  fact  that,  in  spite  of 
the  different  meaning  of  +  ,  developed  functions  in  Boole's  system  and  in 
Jevons's  always  have  the  same  form. 

(11)  The  "logical  alphabet"  is  made  up  of  any  term  plus  its  negative, 
a  +  -a.  It  follows  immediately  from  this  and  law  (10)  that  the  logical 
alphabet  for  any  number  of  terms,  a,  b,  c,  .  .  .  ,  will  be 

(a  +  -a)  (b  +  -b)  (c  +  -c) . . . 

and  will  have  the  character  which  we  have  described.  It  corresponds  to 
the  expansion  of  1  in  Boole's  system  because  it  is  a  developed  function  and 
its  terms  are  mutually  exclusive. 

A  procedure  by  which  Jevons  sets  great  store  is  the  "substitution  of 
similars",  of  a  for  b  or  b  for  a  when  a  =  b.  Not  only  is  this  procedure  valid 
when  the  expressions  in  which  a  and  b  occur  belong  to  the  system,  but  it 
holds  good  whatever  the  rational  complex  in  which  a  and  6  stand.  He 
considers  this  the  first  principle  of  reasoning,  more  fundamental  than 
Aristotle's  dictum  de  omni  et  nullo.ns  In  this  he  is  undoubtedly  correct, 
and  yet  there  is  another  principle,  which  underlies  Aristotle's  dictum,  which 
is  equally  fundamental — the  substitution  for  variables  of  values  of  these 
variables.  And  this  procedure  is  not  reducible  to  any  substitution  of 
equivalents. 

The  only  copulative  relation  in  the  system  is  [.=  ];  hence  the  expression 
of  simple  logical  propositions  is  substantially  the  same  as  with  Boole: 

All  a  is  6:  a  =  a  b 

No  a  is  b :  a  =  a  -b 

Some  a  is  b:    c  a  —  c  ab        or         U  a  =  V  ab 

"U"  is  used  to  suggest  "Unknown". 

The  methods  of  working  with  this  calculus  are  in  some  respects  simpler 
than  Boole's,  in  some  respects  more  cumbersome.  But,  as  Jevons  claims, 
they  are  obvious  while  Boole's  are  not.  Eliminations  are  of  two  sorts, 
"intrinsic"  and  "extrinsic".  Intrinsic  eliminations  may  be  performed  by 
substituting  for  any  part  of  one  member  of  an  equation  the  whole  of  the 
other.  Thus  from  a  =  b  c  d,  we  get 

a  =  a  c  d  =  ab  d  =  ac  =  a  d 

This  rule  follows  from   the  principles  a  a  =  a,   ab  =  b  a,   and  if  a  =  b, 
ac  =  b  c.     For  example 

If      a  =  b  c  d 

a- a  =  bed-bed  =  bb-c  c-dd  =  be  d-d  =  ad. 
118  See  Substitution  of  Similars,  passim. 


76  A  Survey  of  Symbolic  Logic 

Also,  in  cases  where  a  factor  or  a  term  of  the  form  a(b  +  -6),  or  of  the  form 
a  -a,  is  involved,  eliminations  may  be  performed  by  the  rules  a(b  +  -6)  =  a 
and  a  -a  =  0. 

Extrinsic  elimination  is  that  simplification  or  "solution"  of  equations 
which  may  occur  when  two  or  more  are  united.  Jevons  does  not  add  or 
multiply  such  equations  but  uses  them  as  a  basis  for  striking  out  terms  in 
the  same  "logical  alphabet". 

This  method  is  equivalent,  in  terms  of  current  procedures,  to  first 
forming  the  expansion  of  1  (which  contains  the  terms  of  the  logical  alphabet) 
and  then  putting  any  equations  given  in  the  form  in  which  one  member 
is  0  and  "subtracting"  them  from  the  expansion  of  1.  But  Jevons  did  not 
hit  upon  the  current  procedures.  His  own  is  described  thus:  m 

"1.  Any  premises  being  given,  form  a  combination  containing  every 
term  involved  therein.  Change  successively  each  simple  term  of  this 
into  its  contrary  [negative],  so  as  to  form  all  the  possible  combinations  of 
the  simple  terms  and  their  contraries.  [E.  g.,  if  a,  b,  and  c  are  involved, 
form  the  "logical  alphabet"  of  all  the  terms  in  the  expansion  of 

(a  +  -a)  (6  +  -b)  (c  +  -c).] 

"2.  Combine  successively  each  such  combination  [or  term,  as  a  be,] 
with  both  members  of  a  premise.  When  the  combination  forms  a  con- 
tradiction [an  expression  having  a  factor  of  the  form  (a -a)]  with  neither 
side  of  a  premise,  call  it  an  included  subject  of  the  premise;  when  it  forms  a 
contradiction  with  both  sides,  call  it  an  excluded  subject  of  the  premise; 
when  it  forms  a  contradiction  with  one  side  only,  call  it  a  contradictory  com- 
bination or  subject,  and  strike  it  out. 

"We  may  call  an  included  or  excluded  subject  a  possible  subject  as 
distinguished  from  a  contradictory  combination  or  impossible  subject. 

"3.  Perform  the  same  process  with  each  premise.  Then  a  combination 
is  an  included  subject  of  a  series  of  premises,  when  it  is  an  included  subject 
of  any  one;  it  is  a  contradictory  subject  when  it  is  a  contradictory  subject 
of  any  one;  it  is  an  excluded  subject  when  it  is  an  excluded  subject  of 
every  premise. 

"4.  The  expression  of  any  term  [as  a  or  6]  involved  in  the  premises 
consists  of  all  the  included  and  excluded  subjects  containing  the  term, 
treated  as  alternatives  [in  the  relation  +  ]. 

"5.  Such  expressions  may  be  simplified  by  reducing  all  dual  terms  [of 

119  Pure  Logic,  pp.  44—46. 


The  Development  of  Symbolic  Logic  77 

the  form  a(b  +  -b)  ],  and  by  intrinsic  elimination  of  all  terms  not  required 
in  the  expression. 

"  6.  When  it  is  observed  that  the  expression  of  a  term  contains  a  com- 
bination which  would  not  occur  in  the  expression  of  any  contrary  of  that 
term,  we  may  eliminate  the  part  of  the  combination  common  to  the  term 
and  its  expression.  .  .  . 

"7.  Unless  each  term  of  the  premises  and  the  contrary  of  each  appear 
in  one  or  other  of  the  possible  subjects,  the  premises  must  be  deemed  in- 
consistent or  contradictory.  Hence  there  must  always  remain  at  least  two 
possible  subjects. 

"Required  by  the  above  process  the  inferences  of  the  premise  a  =  b  c. 

"The  possible  combinations  of  the  terms  a,  b,  c,  and  their  contraries 
are  as  given  [in  the  column  at  the  left,  which  is,  for  this  case,  the  'logical 
alphabet'].  Each  of  these  being  combined  with  both  sides  of  the  premise, 
we  have  the  following  results: 

ab  c                      ab  c  =  ab  c  abc  included  subject 

ab-c                     ab  -c  =  ab  c  -c  =  0  ab  -c  contradiction 

a-b  c                     a-b  c  =  ab  -b  c  =  0  a-b  c  contradiction 

a  -b  -c                  a-b  -c  =  a  b  -b  c  -c  =  0  a-b  -c  contradiction 

-abc  0  =  a-ab  c  =  -ab  c  -abc  contradiction 

-ab-c  0  =  a  -a  b  -c  —  -ab  c  -c  =  0  -ab-c  excluded  subject 

-a-b  c  0  =  a  -a  -b  c  =  -a  b  -b  c  =0  -a-b  c  excluded  subject 

-a-b  -c  0  =  a  -a  -b  -c  =  -ab  -b  c  -c  =  0  -a  -b  -c  excluded  subject 

" It  appears,  then,  that  the  four  combinations  ab-c  to  -abc  are  to 
be  struck  out,  and  only  the  rest  retained  as  possible  subjects.  Suppose  we 
now  require  an  expression  for  the  term  -b  as  inferred  from  the  premise 
a  =  b  c.  Select  from  the  included  and  excluded  subjects  such  as  contain  -b, 
namely  -a  -b  c  and  -a  -b  -c. 

"  Then  -b  =  -a-b  c  +  -a  -b  -c,  but  as  -a  c  occurs  only  with  -b,  and 
not  with  b,  its  contrary,  we  may,  by  Rule  6,  eliminate  -b  from  -a-b  c; 
hence  -b  =  -a  c  +  -a  -b  -c." 

This  method  resembles  nothing  so  much  as  solution  by  means  of  the 
Venn  diagrams  (to  be  explained  in  Chapter  III).  The  "logical  alphabet" 
is  a  list  of  the  different  compartments  in  such  a  diagram;  those  marked 
"contradiction"  are  the  ones  which  would  be  struck  out  in  the  diagram  by 
transforming  the  equations  given  into  the  form  in  which  one  member  is  0. 


78  A  Survey  of  Symbolic  Logic 

The  advantage  which  Jevons  claims  for  his  method,  apart  from  its  obvious- 
ness,— namely,  that  the  solutions  for  different  terms  do  not  require  to  be 
separately  performed, — is  also  an  advantage  of  the  diagram,  which  exhibits 
all  the  possibilities  at  once. 

If  any  problem  be  worked  out  by  this  method  of  Jevons  and  also  that 
of  Boole,  it  will  be  found  that  the  comparison  is  as  follows:  The  "logical 
alphabet"  consists  of  the  terms  which  when  added  give  1,  or  the  universe. 
Any  term  marked  "contradiction"  will,  by  Boole's  method,  have  the  coef- 
ficient 0  or  1/0;  any  term  marked  "included  subject"  will  have  the  coef- 
ficient 1;  any  marked  "excluded  subject"  will  have  the  coefficient  0/0,  or  v 
where  v  is  arbitrary.  If,  then,  we  remember  that,  according  to  Boole, 
terms  with  the  coefficient  1/0  are  equated  to  0  and  thus  eliminated,  we 
see  that  the  two  methods  give  substantially  the  same  results.  The  single 
important  difference  is  in  Boole's  favor:  the  method  of  Jevons  does  not 
distinguish  decisively  between  the  coefficients  1  and  V.  If,  for  example, 
the  procedure  of  Jevons  gives  x  =  x  -y  z,  Boole's  will  give  either  x  =  -y  z 
or  x  =  v-y  z. 

One  further,  rather  obvious,  principle  may  be  mentioned :  12° 

Any  subject  of  a  proposition  remains  an  included,  excluded,  or  con- 
tradictory subject,  after  combination  with  any  unrelated  terms.  This 
means  simply  that,  in  any  problem,  the  value  of  a  term  remains  its  value 
as  a  factor  when  the  term  is  multiplied  by  any  new  terms  which  may  be 
introduced  into  the  problem.  In  a  problem  involving  a,  b,  and  c,  let 
a  -b  c  be  a  "contradictory"  term.  Then  if  x  be  introduced,  a-bcx  and 
a  -b  c  ~-x  will  be  "contradictory". 

On  the  whole  Jevons's  methods  are  likely  to  be  tedious  and  have  little 
of  mathematical  nicety  about  them.  Suppose,  for  example,  we  have  three 
equations  involving  altogether  six  terms.  The  "logical  alphabet"  will 
consist  of  sixty-four  members,  each  of  which  will  have  to  be  investigated 
separately  for  each  equation,  making  one  hundred  and  ninety-two  separate 
operations.  Jevons  has  emphasized  his  difference  from  Boole  to  the  extent 
of  rejecting  much  that  would  better  have  been  retained.  It  remained  for 
others,  notably  Mrs.  Ladd-Franklin  and  Schroder,  to  accept  Jevons's 
amended  meaning  of  addition  and  its  attendant  advantages,  yet  retain 
Boole's  methods  of  development  and  similar  methods  of  elimination  and 
solution.  But  Jevons  should  have  credit  for  first  noting  the  main  clue  to 
this  simplification — the  laws  a  +  a  =  a  and  a  +  a  b  =  a. 

120  Pure  Logic,  p.  48. 


The  Development  of  Symbolic  Logic  79 

VII.    PEIRCE 

The  contributions  of  C.  S.  Peirce121  to  symbolic  logic  are  more  numerous 
and  varied  than  those  of  any  other  writer — at  least  in  the  nineteenth 
century.  He  understood  how  to  profit  by  the  work  of  his  predecessors, 
Boole  and  De  Morgan,  and  built  upon  their  foundations,  and  he  anticipated 
the  most  important  procedures  of  his  successors  even  when  he  did  not 
work  them  out  himself.  Again  and  again,  one  finds  the  clue  to  the  most 
recent  developments  in  the  writings  of  Peirce.  These  contributions  may 
be  summed  up  under  three  heads:  (1)  He  improved  the  algebra  of  Boole 
by  distinguishing  the  relations  which  are  more  characteristic  of  logical 
classes  (such  as  multiplication  in  Boole's  algebra)  from  the  relations  which 
are  more  closely  related  to  arithmetical  operations  (such  as  subtraction  and 
division  in  Boole).  The  resulting  algebra  has  certain  advantages  over  the 
system  of  Jevons  because  it  retains  the  mathematical  methods  of  develop- 
ment, transformation,  elimination,  and  solution,  and  certain  advantages 
over  the  algebra  of  Boole  because  it  distinguishes  those  operations  and 
relations  which  are  always  interpretable  for  logical  classes.  Also  Peirce 
introduced  the  "illative"  relation,  "is  contained  in",  or  "implies",  into 
symbolic  logic.  (2)  Following  the  researches  of  De  Morgan,  he  made 
marked  advance  in  the  treatment  of  relations  and  relative  terms.  The 
method  of  dealing  with  these  is  made  more  precise  and  "mathematical", 
and  the  laws  which  govern  them  are  related  to  those  of  Boole's  algebra  of 
classes.  Also  the  method  of  treating  "some"  and  "all"  propositions  as 
sums  (2)  and  products  (II)  respectively  of  "propositions"  containing 
variables  was  here  first  introduced.  This  is  the  historic  origin  of  "formal 
implication"  and  all  that  has  been  built  upon  it  in  the  more  recent  develop- 
ment of  the  logic  of  mathematics.  (3)  Like  Leibniz,  he  conceived  symbolic 
logic  to  be  the  science  of  mathematical  form  in  general,  and  did  much  to 
revive  the  sense  of  logistic  proper,  as  we  have  used  that  term.  He  worked 
out  in  detail  the  derivation  of  various  multiple  algebras  from  the  calculus 
of  relatives,  and  he  improved  Boole's  method  of  applying  symbolic  logic  to 
problems  in  probability. 

121  Charles  Saunders  Peirce  (1839-1914),  son  of  Benjamin  Peirce,  the  celebrated 
mathematician,  A.B.  (Harvard,  1859),  B.S.  (Harvard,  1863),  lecturer  in  logic  at  Johns 
Hopkins,  1890-  ?.  For  a  number  of  years,  Peirce  was  engaged  in  statistical  researches 
for  the  U.  S.  Coast  Survey,  and  was  at  one  time  head  of  the  Department  of  Weights  and 
Measures.  His  writings  cover  a  wide  variety  of  topics  in  the  history  of  science,  meta- 
physics, mathematics,  astronomy,  and  chemistry.  According  to  William  James,  his 
articles  on  "Some  Illustrations  of  the  Science  of  Logic",  Pop.  Sci.  Mo.,  1877-78,  are  the 
source  of  pragmatism. 


80  A  Survey  of  Symbolic  Logic 

We  shall  take  up  these  contributions  in  the  order  named. 

The  improvement  of  the  Boolian  algebra  is  set  forth  mainly  in  the 
brief  article,  "On  an  Improvement  in  Boole's  Calculus  of  Logic",122  and  in 
two  papers,  "On  the  Algebra  of  Logic".123 

It  will  be  remembered  that  Boole's  calculus  has  four  operations,  or  rela- 
tions: a  +  b  indicates  the  class  made  up  of  the  two  mutually  exclusive  classes, 
a  and  b;  [  —  ]  is  the  strict  inverse  of  [  +  ],  so  that  if  x  +  b  =  a,  then  x  =  a  —  6; 
ax  b  or  a  b  denotes  the  class  of  those  things  which  are  common  to  a  and  b ; 
and  division  is  the  strict  inverse  of  multiplication,  so  that  if  x  b  =  a,  then 
x  =  a/6.  These  relations  are  not  homogeneous  in  type.  Boole's  [  +  ] 
and  [  — ]  have  properties  which  approximate  closely  those  of  arithmetical 
addition  and  subtraction.  If  [n]x  indicate  the  number  of  members  of  the 
class  x, 

[n]a  +  [n]b  —  [n](a+  6) 

because  a  and  6  are  mutually  exclusive  classes,  and  every  member  of  a 
is  a  member  of  (a  +  6)  and  every  member  of  b  is  a  member  of  (a  +  6).  This 
relation,  then,  differs  from  arithmetical  addition  only  by  the  fact  that 
a  and  b  are  not  necessarily  to  be  regarded  as  numbers  or  quantities.  Simi- 
larly, 

[n]a  —  [n]b  =  [n](a  —  6) 

But  in  contrast  to  this,  for  Boole's  a  x  b  or  a  b, 

[n]a  x  [n]b  =  [n](a  6) 

will  not  hold  except  for  0  and  1 :  this  relation  is  not  of  the  type  of  its  arith- 
metical counterpart.  And  the  same  is  true  of  its  inverse,  a/6.  Thus,  in 
Boole's  calculus,  addition  and  subtraction  are  relations  of  the  same  type 
as  arithmetical  addition  and  subtraction;  but  multiplication  and  division 
are  different  in  type  from  arithmetical  multiplication  and  division. 

Peirce  rounds  out  the  calculus  of  Boole  by  completing  both  sets  of  these 
relations,  adding  multiplication  and  division  of  the  arithmetical  type,  and 
addition  and  subtraction  of  the  non-arithmetical  type.124  The  general 
character  of  these  relations  is  as  follows : 

122  Proc.  Amer.  Acad.,  vu,  250-61.     This  paper  will  be  referred  to  hereafter  as  "Boole's 
Calculus  ". 

123  Amer.  Jour.  Math.,  in  (1880),  15-57,  and  vu  (1885),  180-202.     These  two  papers 
will  be  referred  to  hereafter  as  Alg.  Log.  1880,  and  Alg.  Log.  1885,  respectively. 

124  "Boole's  Calculus,"  pp.  250-54. 


The  Development  of  Symbolic  Logic  81 

A.     The  " '  N on- Arithmetical"  or  Logical  Relations 

(1)  a  +  b  denotes  the  class  of  those  things  which  are  either  a's  or  6's  or 
both.™ 

(2)  The  inverse  of  the  above,  a  \-b,  is  such  that  if  x  +  6  =  a,  then 
x  =  a  \-b. 

Since  x  and  b,  in  x  +  b,  need  not  be  mutually  exclusive  classes,  a  \-  b  is 
an  ambiguous  function.     Suppose  x  +  b  =  a  and  all  6  is  x.     Then 

a  h  b  =  x,        and        a  \-  b  =  a 

Thus  a  (-  b  has  an  upper  limit,  a.  But  suppose  that  x  +  b  —  a  and  no  b 
is  x.  Then  a  \-  b  coincides  with  a  —  b  (a  which  is  not  6) — i.  e., 

a  \-  b  =  x,        and        a  \-  b  =  a  —  b 

Thus  a  \-  b  has  a  lower  limit,  a  —  b,  or  (as  we  elsewhere  symbolize  it) 
a  -b.  And  in  any  case,  a  \-  b  is  not  interpretable  unless  all  6  is  a,  the 
class  b  contained  in  the  class  a.  We  may  summarize  all  these  facts  by 

a  \-  b  =  a  -b  +  v  a  b  +  [0]  -a  b 

where  v  is  undetermined,  and  [0]  indicates  that  the  term  to  which  it  is 
prefixed  must  be  null. 

(3)  a  b  denotes  the  class  of  those  things  which  are  both  a's  and  6's. 
This  is  Boole's  a  b. 

(4)  The  inverse  of  the  preceding,  a/6  such  that  if  6  x  =  a,  then  x  =  a/6. 
This  is  Boole's  a/6. 

a/6  is  an  ambiguous  function.     Its  upper  limit  is  a +  -6;    its  lower 
limit,  a.126     It  is  uninterpretable  unless  6  is  contained  in  a — i.  e., 

a/6  =  a  6  +  v  -a  -6  +  [0]  a  -6 

B.     The  "Arithmetical"  Relations 

(5)  a  +  6  denotes  the  class  of  those  things  which  are  either  a's  or  6's, 
where  a  and  6  are  mutually  exclusive  classes.     This  is  Boole's  a  +  6. 

a  +  6  =  a  -6  +  -a  6  +  [0]  a  6 

(6)  The  inverse  of  the  preceding,     a  —  6  signifies  the  class  "a  which 
is  not  6".     As  has  been  mentioned,  it  coincides  with  the  lower  limit  of  a  h6. 

(7)  a  X  6  and  a  -i-  6  are  strictly  analogous  to  the  corresponding  relations 

125  peirce  indicates  the  logical  relations  by  putting  a  comma  underneath  the  sign  of 
the  relation:  that  which  is  both  a  and  b  is  a,  b. 

126  Peirce  indicates  the  upper  limit  by  a  :  6,  the  lower  limit  by  a  -f-  b.    These  occur 
only  in  the  paper  "Boole's  Calculus". 

7 


82  A  Survey  of  Symbolic  Logic 

of  arithmetic.  They  have  no  such  connection  with  the  corresponding 
"logical"  relations  as  do  a  +  6  and  a  —  b.  Peirce  does  not  use  them 
except  in  applying  this  system  to  probability  theory. 

For  the  "logical"  relations,  the  following  familiar  laws  are  stated:  m 
a+  a  =  a  a  a  =  a 

a  +  b  =  b  +  a  ab  =  b  a 

(a  +  6)  +  c  =  a  +  (b  +  c)         (a  6)c  =  a(b  c) 
(a+b)c  =  ac  +  b  c  a  b  +  c  =  (a  +  c) (b  +  c) 

The  last  two  are  derived  from  those  which  precede. 

Peirce's  discussion  of  transformations  and  solutions  in  this  system  is 
inadequate.  Any  sufficient  account  would  carry  us  quite  beyond  what 
he  has  given  or  suggested,  and  require  our  report  to  be  longer  than  the 
original  paper.  We  shall  be  content  to  suggest  ways  in  which  the  methods 
of  Boole's  calculus  can  be  extended  to  functions  involving  those  relations 
which  do  not  appear  in  Boole.  As  has  been  pointed  out,  if  any  function 
be  developed  by  Boole's  laws, 
/(*)  =/(!)•* +/(())•-*, 

<p(x,  y}  =  <p(\,  l)-xy  +  <p(\,  ty-x-y  +  <p(0,  !)•  -xy  +  <p(Q,  0)-  -x-y, 
Etc.,  etc., 

the  terms  on  the  right-hand  side  of  these  equations  will  always  represent 
mutually  exclusive  classes.  That  is  to  say,  the  difference  between  the 
"logical"  relation,  +  ,  and  the  "arithmetical"  relation,  +,  here  vanishes. 
Thus  any  relation  in  this  system  of  Peirce's  can  be  interpreted  by  developing 
it  according  to  the  above  laws,  provided  that  we  can  interpret  these  rela- 
tions when  they  appear  in  the  coefficients.  And  the  correct  interpretation 
of  these  coefficients  can  always  be  discovered. 
Developing  the  "logical"  sum,  x  +  y,  we  have, 

x  +  y  =  (1  +  1)  -x  y  +  (1  +  0)  -x  -y  +  (0  +  1)  •  -x  y  +  (0  +  0)  •  -x-y 

Comparing  this  with  the  meaning  of  x  +  y  given  above,  we  find  that  (1+1) 
=  1,  (1  +  0)  =  1,  (0+1)  =  1,  and  (0  +  0)  =  0. 

Developing  the  "logical"  difference,  a  \-b,  we  have 

x  \-y  =  (I  |-l)-*yi-(l  hO)-z-2/+(0  hl)--zy+(0  hO)-  -x-y 

Comparing  this  with  the  discussion  of  x  \-y  above,  we  see  that  (1  1-1)  is 
equivalent   to   the   undetermined    coefficient   v;    that    (1  hO)  =  1;    that 
127  "Boole's  Calculus,"  pp.  250-53. 


The  Development  of  Symbolic  Logic  83 

(0  hi)  is  equivalent  to  [0],  which  indicates  that  the  term  to  which  it  is 
prefixed  must  be  null,  and  that  (0  |-0)  =  0. 

The  interpretation  of  the  "arithmetical"  relations,  X  and  H-,  in  coef- 
ficients of  class-symbols  is  not  to  be  attempted.  These  are  of  service  only 
in  probability  theory,  where  the  related  symbols  are  numerical  in  their 
significance. 

The  reader  does  not  require  to  be  told  that  this  system  is  too  complicated 
to  be  entirely  satisfactory.  In  the  "  Description  of  a  Notation  for  the 
Logic  of  Relatives",  all  these  relations  except  -f-  are  retained,  but  in  later 
papers  we  find  only  the  "logical"  relations,  a+  b  and  a  b. 

The  relation  of  "inclusion  in"  or  "being  as  small  as"  (which  we  shall 
symbolize  by  c)128  appears  for  the  first  time  in  the  "Description  of  a 
Notation  for  the  Logic  of  Relatives".129  Aside  from  its  treatment  of 
relative  terms  and  the  use  of  the  "arithmetical"  relations,  this  monograph 
gives  the  laws  of  the  logic  of  classes  almost  identically  as  they  stand  in  the 
algebra  of  logic  today.  The  following  principles  are  stated.130 

(1)  If  x  cy  and  y  cz,  then  x  cz. 

(2)  If  a  c  b,  then  there  is  such  a  term  x  that  a  +  x  =  6. 

(3)  If  acb,  then  there  is  such  a  term  y  that  b  y  =  a. 

(4)  If  b  x  =  a,  then  acb. 

(5)  If  acb,  (c  +  a)c(c  +  6). 

(6)  If  acb,  c  a  ccb. 

(7)  If  a  c  b,  a  c  c  b  c. 

(8)  a  b  c  a. 

(9)  xc(x  +  y). 

(10)  x  +  y  =  y  +  x. 

(11)  (x  +  y)  +  z  =  x  +  (y  +  z). 

(12)  x(y  +  z)  =  xy  +  xz. 

(13)  xy  =  yx. 

(14)  (x  y)z  =  x(yz). 

(15)  xx  =  x. 

(16)  x  -x  =  O.131 

(17)  x  +  -x  =  1. 

128  Peirce's  symbol  is  —  <  which  he  explains  as  meaning  the  same  as  <  but  being  sim- 
pler to  write. 

129  Memoirs  of  the  Amer.  Acad.,  n.  s.,  ix  (1867),  317-78. 

130  "Description  of  a  Notation  for  the  Logic  of  Relatives,"  loc.  tit.,  pp.  334-35,  338-39, 
342. 

131  In  this  paper,  not-x  is  symbolized  by  nx,  "different  from  every  x,"  or  by  a~x. 


84  A  Survey  of  Symbolic  Logic 

(18)  z  +  0  =  x. 

(19)  ar  +  1  =  1. 

(20)  <p(x)  =  <p(l)-x  +  <p(Q)'  -x. 

(21)  «,(*)  =  Ml)  +  *][?(0)  +  ^]. 

(22)  If  9(x)  =0,   ?(1)  •*(())  =0. 

(23)  If  v(x)  =  1,  ?(1)+?(0)  =  1. 

The  last  of  these  gives  the  equation  of  condition  and  the  elimination  re- 
sultant for  equations  with  one  member  1.  Boole  had  stated  (22),  which 
is  the  corresponding  law  for  equations  with  one  member  0,  but  not  (23). 
Most  of  the  above  laws,  beyond  (9),  had  been  stated  either  by  Boole  or 
by  Jevons.  (1)  to  (9)  are,  of  course,  novel,  since  the  relation  c  appears 
here  for  the  same  time  since  Lambert. 

Later  papers  state  further  properties  of  the  relation   c ,  notably, — 

If  x  c  y,  then  -y  c  -x. 

And  the  methods  of  elimination  and  solution  are  given  in  terms  of  this 
relation.132  Also,  these  papers  extend  the  relation  to  propositions.  In  this 
interpretation,  Peirce  reads  x  cy,  "If  a:  is  true,  y  is  true,"  but  he  is  well 
aware  of  the  difference  between  the  meaning  of  x  cy  and  usual  significance 
of  "x  implies  y".  He  says: 133 

"It  is  stated  above  that  this  means  'if  £  is  true,  y  is  true'.  But  this 
meaning  is  greatly  modified  by  the  circumstance  that  only  the  actual  state 
of  things  is  referred  to.  ...  Now  the  peculiarity  of  the  hypothetical 
proposition  [ordinarily  expressed  by  'if  a;  is  true,  y  is  true']  is  that  it  goes 
out  beyond  the  actual  state  of  things  and  declares  what  would  happen  were 
things  other  than  they  are  or  may  be.  The  utility  of  this  is  that  it  puts 
us  in  possession  of  a  rule,  say  that '  if  A  is  true,  B  is  true ',  such  that  should 
we  afterward  learn  something  of  which  we  are  now  ignorant,  namely  that 
A  is  true,  then,  by  virtue  of  this  rule,  we  shall  find  that  we  know  something 
else,  namely,  that  B  is  true.  [In  contrast  to  this]  .  .  .  the  proposition, 
a  c  b,  is  true  if  a  is  false  or  if  b  is  true,  but  is  false  if  a  is  true  while  b  is  false. 
.  .  .  For  example,  we  shall  see  that  from  -(x  cy)  [the  negation  of  x  cy] 
we  can  infer  zcx.  This  does  not  mean  that  because  in  the  actual  state  of 
things  x  is  true  and  y  false,  therefore  in  every  state  of  things  either  z  is 
false  or  x  true ;  but  it  does  mean  that  in  whatever  state  of  things  we  find  x 
true  and  y  false,  in  that  state  of  things  either  z  is  false  or  x  is  true  [since, 
ex  hypothesi,  x  is  true  anyway]." 

132  Alg.  Log.  1880,  see  esp.  §  2. 
*»Alg.  Log.  1885,  pp.  186-87. 


The  Development  of  Symbolic  'Logic  85 

We  now  call  this  relation,  x  c  y,  "  material  implication, "  and  the  peculiar 
theorems  which  are  true  of  it  are  pretty  well  known.  Peirce  gives  a  number 
of  them.  They  will  be  intelligible  if  the  reader  remember  that  x  c  y  means, 
"The  actual  state  of  things  is  not  one  in  which  x  is  true  and  y  false". 

(1)  xc(ycx).     This  is  the  familiar  theorem:    "A  true  proposition  is 
implied  by  any  proposition". 

(2)  [(x  cy)  ex]  ex.     If  "x  implies  y"  implies  that  x  is  true,  then  x  is 
true. 

(3)  [(x  c  y)  c  a]  c  x,  where  a  is  used  in  such  a  sense  that  (x  c  y)  c  a 
means  that  from  x  c  y  every  proposition  follows. 

The  difference  between  "material  implication"  and  the  more  usual 
meaning  of  "implies"  is  a  difficult  topic  into  which  we  need  not  go  at  this 
time.134  But  it  is  interesting  to  note  that  Peirce,  who  introduced  the 
relation,  understood  its  limitations  as  some  of  his  successors  have  not. 

Other  theorems  in  terms  of  this  relation  are : 

(4)  x  ex. 

(5)  [xc(ycz)]c  [yc(xcz)]. 

(6)  xc[(xcy)cy]. 

(7)  (xcy)  c  [(y  cz)  c(xcz)].     This  is  a  fundamental  law,  since  called 
the  "Principle  of  the  Syllogism". 

Peirce  worked  most  extensively  with  the  logic  of  relatives.  His  interest 
here  reflects  a  sense  of  the  importance  of  relative  terms  in  the  analyses  of 
mathematics,  and  he  anticipates  to  some  extent  the  methods  of  such  later 
researches  as  those  of  Peano  and  of  Principia  Mathematica.  To  follow 
his  successive  papers  on  this  topic  would  probably  result  in  complete  con- 
fusion for  the  reader.  Instead,  we  shall  make  three  divisions  of  this  entire 
subject  as  treated  by  Peirce:  (1)  the  modification  and  extension  of  De 
Morgan's  calculus  of  relatives  by  the  introduction  of  a  more  "mathe- 
matical" symbolism — for  the  most  part  contained  in  the  early  paper, 
"Description  of  a  Notation  for  the  Logic  of  Relatives";  (2)  the  calculus 
of  relations,  expressed  without  the  use  of  exponents  and  in  a  form  which 
makes  it  an  extension  of  the  Boolean  algebra — a  latef  development  which 
may  be  seen  at  its  best  in  "The  Logic  of  Relatives",  Note  B  in  the  Studies 
in  Logic  by  members  of  Johns  Hopkins  University;  and  (3)  the  systematic 
consideration  of  the  theory  of  relatives,  which  is  scattered  throughout  the 
papers,  but  has  almost  complete  continuity. 

134  But  see  below,  Chap,  iv,  Sect,  i,  and  Chap,  v,  Sect.  v. 


86  A  Survey  of  Symbolic  Logic 

The  terms  of  the  algebra  of  relatives  may  usually  be  regarded  as  simple 
relative  terms,  such  as  "ancestor",  "lover,"  etc.  Since  they  are  also  class 
names,  they  will  obey  all  the  laws  of  the  logic  of  classes,  which  may  be 
taken  for  granted  without  further  discussion.  But  relative  terms  have 
additional  properties  which  do  not  belong  to  non-relatives;  and  it  is  to 
these  that  our  attention  must  be  given.  If  w  signifies  "woman"  and  *, 
"servant,"  logic  is  concerned  not  only  with  such  relations  as  s  w,  the 
"logical  product"  "servant  woman",  s  +  w,  the  "logical  sum"  "either 
servant  or  woman  (or  both)",  and  s  cw,  "the  class  'servants'  is  contained 
in  the  class  'women',  —  relations  which  belong  to  class-terms  in  general— 
but  also  with  the  relations  first  symbolized  by  De  Morgan,  "servant  of  a 
woman,"  "servant  of  every  woman,"  and  "servant  of  none  but  women". 

We  may  represent  "servant  of  a  woman"  by  s\w.m  This  is  a  kind  of 
"multiplication"  relation.  It  is  associative, 

8\(1\W)    =   (S\1)\W 

"Servant  of  a  lover-of-a-woman  "  is  "  servant-of-a-lover  of  a  woman". 
Also,  it  is  distributive  with  respect  to  the  non-relative  "addition"  symbol- 
ized by  +  , 


s 


=  sm 


w 


"Servant  of  either  a  man  or  a  woman"  is  "servant  of  a  man  or  servant  of 
a  woman  ".  But  it  is  not  commutative  :  8\lis  not  l\s,  "  servant  of  a  lover  " 
is  not  equivalent  to  "lover  of  a  servant".  To  distinguish  s  w  from  s  w, 
or  s  x  w  —  the  class  of  those  who  are  both  servants  and  women  —  we  shall 
call  s  w  the  relative  product  of  s  and  w. 

For  "servant  of  every  woman"  Peirce  proposed  sw,  and  for  "servant 
of  none  but  women"  *w.  As  we  shall  see,  this  notation  is  suggested  by 
certain  mathematical  analogies.  We  may  represent  individual  members 
of  the  class  w  as  Wi,  Wz,  W3,  etc.,  and  the  class  of  all  the  Ws  as  W\  +  W2 
+  Wz  +  .  .  .  .  Remembering  the  interpretation  of  +  ,  we  may  write 

w  =  Wl  +  W2  +  W3  +  .  .  . 

and  this  means,  "The  class-term,  w,  denotes  W\  or  Wz  or  W3  or  ...," 
that  is,  w  denotes  an  unspecified  member  of  the  class  of  Ws.  The  servant 
of  a  (some,  any)  woman  is,  then,  s  w. 

s  w  =  8\(Wi+Wt+W>+  ...)  =  s  Wi  +  s^z  +  s  W3+  ... 

"A  woman"  is  either  W  \  or  W2  or  W3,  etc.;  "servant  of  a  woman"  is  either 
136  Peirce's  notation  for  this  is  s  w;  he  uses  s,  w  for  the  simple  logical  product. 


The  Development  of  Symbolic  Logic  87 

servant  of  Wi  or  servant  of  Wz  or  servant  of  W3,  etc.  Similarly,  "  servant 
of  every  woman"  is  servant  of  W\  and  servant  of  W2  and  servant  of  W3, 
etc.  ;  or  remembering  the  interpretation  of  x  , 


where,  of  course,  s\  Wn  represents  the  relative  product,  "s  of  Wn,"  and  x 
represents  the  non-relative  logical  product  translated  by  "and".  The 
above  can  be  more  briefly  symbolized,  following  the  obvious  mathematical 
analogies, 

w  =  VW 

w  =  2w(s  W) 


Unless  w  represent  a  null  class,  we  shall  have 


or        swcs  w 

The  class  "servants  of  every  woman"  is  contained  in  the  class  "servants 
of  a  woman".     This  law  has  numerous  consequences,  some  of  which  are: 

(l\8)»c(l\8  w} 

A  lover  of  a  servant  of  all  women  is  a  lover  of  a  servant  of  a  woman. 

J«wc  (J|*)« 

A  lover  of  every  servant-of-all-women  stands  to  every  woman  in  the  rela- 
tion of  lover-of-a-servant  of  hers  (unless  the  class  sw  be  null). 


Is  I  «  c  Is 


w 


A  lover  of  every  servant-of-a-woman  stands  to  a  (some)  woman  in  the 
relation  of  lover-of-a-servant  of  hers. 
From  the  general  principle,136 


134  The  proof  of  this  theorem  is  as  follows: 

a  =  ab  c  ...  +ab  -c  ...  +a  -b  c  ...  +  ..., 
or  a  =  a  b  c  .  .  .  +  P,  where  P  is  the  sum  of  the  remaining  terms. 
Whence,  if  O  represent  any  relation  distributive  with  respect  to  +  , 

mOa  =  mOa  be  ...  +mOP 

Similarly,  mQb  =  mOa  be...  +mQQ 

mQc  =  mOab  c  .  .  .  +mQR 
Etc.,  etc. 

Now  let  o,  b,  c,  etc.,  be  respectively  /(xi),  /(%),  /(a;3),  etc.,  and  multiply  together  all 
the  above  equations.    On  the  left  side,  we  have 


88  A  Survey  of  Symbolic  Logic 

we  have  also, 


*wc*w,        or          *«c*» 

A  lover  of  a  (some)  servant-of-every-woman  stands  to  every  woman  in  the 
relation  of  lover-of-a-servant  of  hers. 

We  have  also  the  general  formulae  of  inclusion, 

If  Ics,    then  lwcsw 
and,       If  s  c  w,  then  lw  c  I" 

The  first  of  these  means:  If  all  lovers  are  servants,  then  a  lover  of  every 
woman  is  also  a  servant  of  every  woman.  The  second  means  :  If  all  servants 
are  women,  then  a  lover  of  every  woman  is  also  a  lover  of  every  servant. 
These  laws  are,  of  course,  general.  We  have  also:  137 

(l\8)\W  =   l\(8   W) 

(/•)«  =  ;<•!») 

l*+w    —    Is  x  lw 

The  last  of  these  is  read:  A  lover  of  every  person  who  is  either  a  servant 
or  a  woman  is  a  lover  of  every  servant  and  a  lover  of  every  woman.  An 
interesting  law  which  remainds  us  of  Lambert's  "Newtonian  formula"  is, 


One  who  is  either-lover-or-servant  of  every  woman,  is  either  lover  of  every 
woman  or,  for  some  portion  q  of  the  class  women,  is  lover  of  every  woman 
except  members  of  q  and  servant  of  every  member  of  q,  or,  finally,  is  servant 
of  every  woman.  Peirce  also  gives  this  law  in  a  form  which  approximates 
even  more  closely  the  binomial  theorem.  The  corresponding  law  for  the 
product  is  simpler, 

(lxs)w  =  lw  xsw 
which  is 


On  the  right  side,  we  have 

(mOabc  .  .  .)  +  (mOP)  +  (mOQ)  +  (mOR)  +  .  .  .,        or         (mOa  b  c  .  .  .)  +K 
where  K  is  a  sum  of  other  terms. 

But  (mQa  b  c  .  .  .  )  is  mO  [/(zO  x  /(x2)  x  /(z3)  .  .  .  ],  which  is 

mOnx/(z) 

Hence  [w  OH  /(«)]+  K  =  Ux[mOf(x)]. 

Hence  mOn/(x)  cUx[mOf(x)]. 

Peirce  does  not  prove  this  theorem,  but  illustrates  it  briefly  for  logical  multiplication  (see 
"Description  of  a  Notation",  p.  346). 

137  <<  Description  of  a  Notation,  p.  334. 


The  Development  of  Symbolic  Logic  89 

One  who  is  both-lover-and-servant  of  every  woman,  is  both  a  lover  of  every 
woman  and  a  servant  of  every  woman. 

Peirce  introduces  a  fourth  term,  and  summarizes  in  a  diagram  the  inclu- 
sion relations  obtained  by  extending  the  formulae  already  given.138  The 
number  of  such  inclusions,  for  four  relatives,  is  somewhat  more  than  one 
hundred  eighty.  He  challenges  the  reader  to  accomplish  the  precise 
formulation  of  these  by  means  of  ordinary  language  and  formal  logic. 

An  s  of  none  but  members  of  w,  Peirce  symbolizes  by  sw.  He  calls  this 
operation  "backward  involution",  and  relatives  of  the  type  *w  he  refers  to 
as  "infinitesimal  relatives",  on  account  of  an  extended  and  difficult  mathe- 
matical analogy  which  he  presents.139  The  laws  of  this  relation  are  analo- 
gous to  those  of  sw. 

If  s  c  w,  then  ls  c  lw 

If  all  servants  are  women,  then  a  lover  of  none  but  servants  is  lover  of  none 
but  women. 

If  I  c  s,  then  sw  c  lw 

If  all  lovers  are  servants,  then  a  servant  of  none  but  women  is  a  lover  of 
none  but  women. 

i(*w)  =  ("*>w 

The  lovers  of  none  but  servants-of-none-but-women  are  the  lovers-of- 
servants  of  none  but  women. 

l+sw  =  lw  x  sw 

Those  who  are  either-lovers-or-servants  of  none  but  women  are  those  who 

are  lovers  of  none  but  women  and  servants  of  none  but  women. 

•  i 
*(w  x  v)  =  aw  x  8v 

The  servants  of  none  but  those  who  are  both  women  and  violinists  are 
those  who  are  servants  of  none  but. women  and  servants  of  none  but  vi©7 
linists. 

{l\a)w  C  (l")w 

Whoever  is  lover-of-a-servant  of  none  but  women  is  a  lover-of-every- 
servant  of  none  but  women. 

l\sw  c  <*Pw 

A  lover  of  one  who  is  servant  to  none  but  women  is  a  lover-of-none-but- 
servants  to  none  but  women. 

ls  w  c  l(s\w) 

138  Ibid.,  p.  347. 

139  Ibid.,  pp.  348  jf. 


90  A  Survey  of  Symbolic  Logic 

Whoever  stands  to  a  woman  in  the  relation  of  lover-of-nothing-but-servants 
of  hers  is  a  lover  of  nothing  but  servants  of  women. 

The  two  kinds  of  involution  are  connected  by  the  laws  : 

'(,«)  =  (',)« 

A  lover  of  none  but  those  who  are  servants  of  every  woman  is  the  same  as 
one  who  stands  to  every  woman  in  the  relation  of  a  lover  of  none  but 
servants  of  hers. 

is  =  -I  -•  14° 

Lover  of  none  but  servants  is  non-lover  of  every  non-servant.  It  appears 
from  this  last  that  x  and  xy  are  connected  through  negation  : 

-(/•)  =  -l\s,     Not  a  lover  of  every  servant  is  non-lover  of  a  servant. 

-('$)  =  l\-s,  Not  a  lover  of  none  but  servants  is  lover  of  a  non- 
servant. 

l-s  =  ~(l\s)  =  -Is,  A  lover  of  none  but  non-servants  is  one  who  is 
not  lover-of-a-servant,  a  non-lover  of  every  servant. 

~ls  =  -(-l\-s)  =  l~s,  A  non-lover  of  none  but  servants  is  one  who  is 
not  a  non-lover-of-a-non-servant,  a  lover  of  every  non-servant. 

We  have  the  further  laws  governing  negatives  :  141 

-[(Zx*)»]  =  -(*»)+  -(*<•) 


-(£•+«)  =  -(/•)  +  -(/<«) 

In  the  early  paper,  "On  the  Description  of  a  Notation  for  the  Logic 
of  Relatives",  negatives  are  treated  in  a  curious  fashion.  A  symbol  is 
used  for  "different  from"  and  the  negative  of  s  is  represented  by  tts,  "differ- 
ent from  every  s".  Converses  are  barely  mentioned  in  this  study.  In  the 
paper  of  1880,  converses  and  negatives  appear  in  their  usual  notation, 
"relative  addition"  is  brought  in  to  balance  "relative  multiplication",  and 
the  two  kinds  of  involution  are  retained.  But  in  "The  Logic  of  Relatives" 
in  the  Johns  Hopkins  Studies  in  Logic,  published  in  1882,  involution  has 
disappeared,  converses  and  negatives  and  "relative  addition"  are  retained. 
This  last  represents  the  final  form  of  Peirce's  calculus  of  relatives.  We 
have  here, 

(1)  Relative  terms,  a,  6,  ...  x,  y,  z. 

(2)  The  negative  of  x,  -x. 

140  See  ibid.,  p.  353.     Not-x  is  here  symbolized  by  (1  —  x). 
"i  Alg.  Log.  1880,  p.  55. 


The  Development  of  Symbolic  Logic  91 

(3)  The  converse  of  x,  ^x.     If  x  is  "lover",  ^x  is  "beloved";  if  KE  is 
"lover",  z  is  "beloved". 

(4)  Non-relative  addition,  a  +  b,  "either  a  or  6". 

(5)  Non-relative  multiplication,  a  x6,  or  a  b,  "both  a  and  b". 

(6)  Relative  multiplication,  a\b,  "a  of  a  b". 

(7)  Relative  addition,  a  t  b,  "a  of  everything  but  6's,  a  of  every  non-6  ". 

(8)  The  relations  =  and  c ,  as  before. 

(9)  The  universal  relation,   1,   "consistent  with,"  which  pairs  every 
term  with  itself  and  with  every  other. 

(10)  The  null-relation,  0,  the  negative  of  1. 

(11)  The  relation  "identical  with",  I,  which  pairs  every  term  with 
itself. 

(12)  The  relation   "different  from",   N,  which  pairs  any  term  with 
every  other  term  which  is  distinct.142 

In  terms  of  these,  the  fundamental  laws  of  the  calculus,  in  addition  to 
those  which  hold  for  class-terms  in  general,  are  as  follows : 

(1)  w(«a)  =  a 

(2)  -(.«)  =  .(-a) 

(3)  (acfe)  =  (w&cwa) 

(4)  If  acb,  then  (a  x)  c(b\x)  and  (x  a)  c(x\b). 

(5)  If  a  c  b,  then  (a  t  x}  c  (b  t  x)  and  (x  t  a)  c  (x  t  6). 

(6)  x  (a 1 6)  =  (x  a)\b 

(7)  x  t  (a  1 6)  =  (x  t  a)  t  b 

(8)  z|(at&)c(z  a)  t& 

(9)  (at  6)  a;  cot  (6  a;) 

(10)  (a  x)  +  (b\x)c(a  +  b)\x 

(11)  z|(at&)c(zta)(zt&) 

(12)  (a+b)xc(a  x)  +  (b\x) 

(13)  (ataO(&tz)c(a|6)  tz 

(14)  -(at 6)  =  -a | -6 

(15)  -(a 1 6)  =  -at -6 

(16)  ^(a  +  6)  =  v/a  +  «6 

(17)  «(a  6)  =  «a  «6 

(18)  w(a  1 6)  =  «a  t  «6 

(19)  w(0|6)  =  «a  v& 

For  the  relations  1,  0,  7,  and  AT,  the  following  additional  formulae  are 
given : 

142 1  have  altered  Peirce's  notation,  as  the  reader  may  see  by  comparison. 


92  A  Survey  of  Symbolic  Logic 

(20)  Oca;  (21)  zcl 

(22)  x  +  Q  =  x  (23)  x-1  =  x 

(24)  x  +  1  =  1  (25)  x-0  =  0 

(26)  x  t 1  =  1  (27)  x  0  =  0 

(28)  1  t  x  =  1  (29)  0  x  =  0 

(30)  x\  N  =  x  (31)  x\I  =  x 

(32)  N  t  x  =  x  (33)  I  x  =  x 

(34)  x  +  -x  =  1  (35)  x  -x  =  0 

(36)  7c[ztv(-aO]  (37)  [z  |  «(-*)]  c  AT 

This  calculus  is,  as  Peirce  says,  highly  multiform,  and  no  general  prin- 
ciples of  solution  and  elimination  can  be  laid  down.143  Not  only  the  variety 
of  relations,  but  the  lack  of  symmetry  between  relative  multiplication  and 
relative  addition,  e.  g.,  in  (10)-(13)  above,  contributes  to  this  multiformity. 
But,  as  we  now  know,  the  chief  value  of  any  calculus  of  relatives  is  not  in 
any  elimination  or  solution  of  the  algebraic  type,  but  in  deductions  to  be 
made  directly  from  its  formulae.  Peirce's  devices  for  solution  are,  there- 
fore, of  much  less  importance  than  is  the  theoretic  foundation  upon  which 
his  calculus  of  relatives  is  built.  It  is  this  which  has  proved  useful  in  later 
research  and  has  been  made  the  basis  of  valuable  additions  to  logistic 
development. 

This  theory  is  practically  unmodified  throughout  the  papers  dealing 
with  relatives,  as  a  comparison  of  "  Description  of  a  Notation  for  the  Logic 
of  Relatives"  with  "The  Logic  of  Relatives"  in  the  Johns  Hopkins  studies 
and  with  the  paper  of  1884  will  indicate. 

"Individual"  or  "elementary"  relatives  are  the  pairs  (or  triads,  etc.) 
of  individual  things.  If  the  objects  in  the  universe  of  discourse  be  A,  B,  C: 
etc.,  then  the  individual  relatives  will  constitute  the  two-dimensional  array, 

A  :  A,  A  :  B,  A  :  C,  A  :  D,   ... 

B  :  A,  B  :  B,  B  :  C,  B  :  D,   ... 

C:A,   C  :B,  C  :  C,  C  :D,    ... 
. . .  Etc.,  etc. 

It  will  be  noted  that  any  individual  thing  coupled  with  itself  is  an  individual 
relative  but  that  in  general  A  :  B  differs  from  B  :  A — individual  relatives 
are  ordered  couples. 

A  general  relative  is  conceived  as  an  aggregate  or  logical  sum  of  such 

143  "Logic  of  Relatives"  in  Studies  in  Logic  by  members  of  Johns  Hopkins  University,. 
p.  193. 


The  Development  of  Symbolic  Logic  93 

individual  relatives.     If  b  represent  "benefactor",  then 

b  =  ZiZyPM/  : «/), 

where  (&),-/  is  a  numerical  coefficient  whose  value  is  1  in  case  I  is  a  bene- 
factor of  J,  and  otherwise  0,  and  where  the  sums  are  to  be  taken  for  all  the 
individuals  in  the  universe.  That  is  to  say,  b  is  the  logical  sum  of  all  the 
benefactor-benefitted  pairs  in  the  universe.  This  is  the  first  formulation 
of  "definition  in  extension",  now  widely  used  in  logistic,  though  seldom  in 
exactly  this  form.  By  this  definition,  b  is  the  aggregate  of  all  the  individual 
relatives  in  our  two-dimensional  array  which  do  not  drop  out  through  having 
the  coefficient  0.  It  is  some  expression  of  the  form, 

b  =  (X:  F)i+(Z:  Y)2+ (X  :  F),+  ... 

If,  now,  we  consider  the  logical  meaning  of  +  ,  we  see  that  this  may  be  read, 
"b  is  either  (X  :  F)i  or  (X  :  F)2  or  (X  :  7)3  or  . . .  ".  To  say  that  b  repre- 
sents the  class  of  benefactor-benefitted  couples  is,  then,  inexact:  b  repre- 
sents an  unspecified  individual  relative,  any  one  of  this  class.  (That  it 
should  represent  "some"  in  a  sense  which  denotes  more  than  one  at  once — 
which  the  meaning  of  +  in  the  general  case  admits — is  precluded  by  the 
fact  that  any  two  distinct  individual  relatives  are  ipso  facto  mutually 
exclusive.)  A  general  relative,  so  defined,  is  what  Mr.  Russell  calls  a 
"real  variable".  Peirce  discusses  the  idea  of  such  a  variable  in  a  most 
illuminating  fashion.144 

"Demonstration  of  the  sort  called  mathematical  is  founded  on  suppo- 
sition of  particular  cases.  The  geometrician  draws  a  figure;  the  algebraist 
assumes  a  letter  to  signify  a  certain  quantity  fulfilling  the  required  condi- 
tions. But  while  the  mathematician  supposes  a  particular  case,  his  hypoth- 
esis is  yet  perfectly  general,  because  he  considers  no  characters  of  the 
individual  case  but  those  which  must  belong  to  every  such  case.  The  ad- 
vantage of  his  procedure  lies  in  the  fact  that  the  logical  laws  of  individual 
terms  are  simpler  than  those  which  relate  to  general  terms,  because  indi- 
viduals are  either  identical  or  mutually  exclusive,  and  cannot  intersect  or 
be  subordinated  to  one  another  as  classes  can.  .  .  . 

"The  old  logics  distinguish  between  individuum  signatum  and  indi- 
mduum  vagum.  'Julius  Caesar'  is  an  example  of  the  former;  'a  certain 
man',  of  the  latter.  The  indimduum  vagum,  in  the  days  when  such  con- 
ceptions were  exactly  investigated,  occasioned  great  difficulty  from  its 
having  a  certain  generality,  being  capable,  apparently,  of  logical  division. 

"*  "Description  of  a  Notation,  pp.  342-44. 


94  A  Survey  of  Symbolic  Logic 

If  we  include  under  indimduum  vagum  such  a  term  as  'any  individual 
man',  these  difficulties  appear  in  a  strong  light,  for  what  is  true  of  any 
individual  man  is  true  of  all  men.  Such  a  term  is  in  one  sense  not  an 
individual  term;  for  it  represents  every  man.  But  it  represents  each  man 
as  capable  of  being  denoted  by  a  term  which  is  individual;  and  so,  though 
it  is  not  itself  an  individual  term,  it  stands  for  any  one  of  a  class  of  such 
terms.  .  .  .  The  letters  which  the  mathematician  uses  (whether  in  algebra 
or  in  geometry)  are  such  individuals  by  second  intention.  .  .  .  All  the 
formal  logical  laws  relating  to  individuals  will  hold  good  of  such  individuals 
by  second  intention,  and  at  the  same  time  a  universal  proposition  may  be 
substituted  for  a  proposition  about  such  an  individual,  for  nothing  can  be 
predicated  of  such  an  individual  which  cannot  be  predicated  of  the  whole 
class." 

The  relative  b,  denoting  ambiguously  any  one  of  the  benefactor-bene- 
fitted  pairs  in  the  universe,  is  such  an  individual  by  second  intention. 
It  is  defined  by  means  of  the  "prepositional  function",  "/  benefits  J", 
as  the  logical  sum  of  the  (/  :  J)  couples  for  which  "I  benefits  J"  is  true. 
The  compound  relations  of  the  calculus  can  be  similarly  defined. 

If  a  =  2<2y(a)iy(7  :  J),  and  b  =  2i2y(6)<y(7  :  J), 
then  a+  b  =  2i2y[(a)i,-  +  (&)</](/  :  J) 

That  is,  if  "agent"  is  the  logical  sum  of  all  the  (7  :  J)  couples  for  which 
"7  is  agent  of  J"  is  true,  and  "benefactor"  is  the  sum  of  all  the  (7  :  J) 
couples  for  which  "I  benefits  J"  is  true,  then  "either  agent  or  benefactor" 
is  the  logical  sum  of  all  the  (7  :  J)  couples  for  which  "  Either  7  is  agent  of 
J  or  7  benefits  J"  is  true.  We  might  indicate  the  same  facts  more  simply 
by  defining  only  the  " propositional  function",  (a  +  6)i,.145 

(a  +  &)t-y  =  (a)  */ +(&).'/ 

The  definition  of  a  +  b  given  above,  follows  immediately  from  this  simpler 
equation.  The  definitions  of  the  other  compound  relations  are  similar: 

(a  x  &),-,•  =  (a)i/x(6)iy 
or        a  x&  =  S<2y[(a)iy  x  (fe)»y](/  :  J) 

"Both  agent  and  benefactor"  is  the  logical  sum  of  the  (7  :  J)  couples  for 
which  "7  is  agent  of  J  and  7  is  benefactor  of  J"  is  true. 

(a  1 6),,  =  2»{(o)ttx(6)w} 
or        a\b  =  2i2y[2*{(a)«  x(6)w}](7  :  J) 
148  See  "Logic  of  Relatives  ",  toe.  dt.t  p.  188. 


The  Development  of  Symbolic  Logic  95 

"Agent  of  a  benefactor"  is  the  logical  sum  of  all  the  (7  :  J)  couples  such 
that,  for  some  H,  "I  is  agent  of  H  and  H  is  benefactor  of  J"  is  true. 

There  are  two  difficulties  in  the  comprehension  of  this  last.  The  first 
concerns  the  meaning  of  "agent  of  a  benefactor".  Peirce,  like  De  Morgan, 
treats  his  relatives  as  denoting  ambiguously  either  the  relation  itself  or 
the  things  which  have  the  relation  —  either  relations  or  relative  terms. 
a  is  either  the  relation  "agent  of"  or  the  class  name  "agent".  Now  note 
that  the  class  name  denotes  the  first  term  in  the  pairs  which  have  the 
relation.  With  this  in  mind,  the  compound  relation,  a\b,  will  become 
clear.  "Agent  of  a  benefactor"  names  the  7's  in  the  I  :  J  pairs  which 
make  up  the  field  of  the  relation,  "  agent  of  a  benefactor  of  ".  Any  reference 
to  the  J's  at  the  other  end  of  the  relation  is  gone,  just  as  "agent"  omits 
any  reference  to  the  J's  in  the  field  of  the  relation  "agent  of".  The  second 
difficulty  concerns  the  operator,  S&,  which  we  have  read,  "For  some  H". 
Consider  any  statement  involving  a  "prepositional  function",  <pz,  where  z 
is  the  variable  representing  the  individual  of  which  <p  is  asserted. 


That  is,  2z<f>z  symbolizes  "Either  <p  is  true  of  Zi  or  <p  is  true  of  Z2  or  <p  is 
true  of  Z3  or  ...  ",  and  this  is  most  simply  expressed  by  "For  some  z  (some 
2  or  other),  <pz".  In  the  particular  case  in  hand,  <pz  is  (a)ih  x  (b}hj,  "I  is 
agent  of  H  and  H  is  benefactor  of  J".  The  terms,  /  and  J,  which  stand  in 
the  relation  "7  is  agent  of  a  benefactor  of  J",  are  those  for  which  there  is 
some  H  or  other  such  that  I  is  agent  of  H  and  H  is  benefactor  of  J. 

Suppose  we  consider  any  "prepositional  function",  tpz  with  the  oper- 
ator n. 

Hz(f>z  =  ipZi  x  ifiZz  x  <pZs  x  ... 

That  is,  nz  <pz  symbolizes  "  <p  is  true  of  Zi  and  <?  is  true  of  Z2  and  <p  is  true 
of  Z3  and  .  .  .  ",  or  "  <p  is  true  for  every  2".  This  operator  is  needed  in  the 
definition  of  a  1  6. 

(at  ft),-/  =  HA  {(a),  -A  +(6)  *,-} 

"7  is  agent  of  everyone  but  benefactors  of  J"  is  equivalent  to  "For  every  H, 
either  7  is  agent  of  7f  or  7T  is  benefactor  of  J". 

at&  =  ZiSnAatt+6 


"Agent  of  all  non-benefactors"  is  the  logical  sum  of  all  the  (7  :  J)  couples 
such  that,  for  every  H,  either  7  is  agent  of  H  or  H  is  benefactor  of  J.  The 
same  considerations  about  the  ambiguity  of  relatives  —  denoting  either  the 


96  A  Survey  of  Symbolic  Logic 

relation  itself  or  those  things  which  are  first  terms  of  the  relation — applies 
in  this  case  also.  We  need  not,  for  the  relations  still  to  be  discussed,  con- 
sider the  step  from  the  definition  of  the  compound  "  prepositional  func- 
tions", (a  t  &),,  in  the  above,  to  the  definition  of  the  corresponding  relation, 
a  t  b.  This  step  is  always  taken  in  exactly  the  same  way. 

The  converse,  converse  of  the  negative,  and  negative  of  the  converse, 
are  very  simply  defined. 

("&);/  =  (&)/< 
[«(-&)],/  =  (-6),< 

[-(<•*)]./  =  -(&),* 

That  the  negative  of  the  converse  is  the  converse  of  the  negative  follows 
from  the  obvious  fact  that  -(&)»/  =  (-&)/». 

All  the  formulae  of  the  calculus  of  relatives,  beyond  those  which  belong 
also  to  the  calculus  of  non-relative  terms,146  may  be  proved  from  such 
definitions.  For  example: 

To  prove,  «(a  +  6)  =  ^a  +  ^b 

v(a  +  b)ij  =  (a +  &),-,-  =  (a),-.-+ (&)/» 
But  (a),-,-  =  (va)a,  and  (&),•»  =  O&)i,- 
Hence  «(a  +  6)t,-  =  (^a)a+  (^b)a 

Hence  St-S/{w(a  +  fe) ,-,-}(/  :  J)  =  SiS/{(wa)t-,-  +  («6) »,-}(/  : «/)  Q.E.D. 
For  the  complete  development  of  this  theory,  there  must  be  a  discussion 
of  the  laws  which  govern  such  expressions  as  (a),-/,  or  in  general,  expressions 
of  the  form  <px,  where  <px  is  a  statement  which  involves  a  variable,  x,  and  <px 
is  either  true  or  false  whenever  any  individual  value  of  the  variable  is 
specified.  Such  expressions  are  now  called  "prepositional  functions".147 
(a),-,  or  in  the  more  convenient  notation,  <px,  is  a  propositional  function  of 
one  variable;  (a)*,-,  or  <f>(x,  y},  may  be  regarded  as  a  propositional  function 
of  two  variables,  or  as  a  function  of  the  single  variable,  the  individual  rela- 
tive (/  :  </),  or  (X  :  Y). 

This  theory  of  propositional  functions  is  stated  in  the  paper  of  1885, 
'On  the  Algebra  of  Logic".  It  is  assumed,  as  also  in  earlier  papers,  that 
he  laws  of  the  algebra  of  classes  hold  for  propositions  as  well.148  The 
dditional  law  which  propositions  obey  is  stated  here  for  the  first  time. 

146  The  formulae  of  the  calculus  of  classes  can  also  be  derived  from  these,  considered  as 
themselves  laws  of  the  calculus  of  propositions  (see  below,  Chap,  vi,  Sec.  iv). 

147  Peirce  has  no  name  for  such  expressions,  though  he  discusses  their  properties  acutely 
(see  Alg.  Log.  1880,  §  2). 

148  This  assumption  first  appears  in  Alg.  Log.  1880. 


The  Development  of  Symbolic  Logic  97 

The  current  form  of  this  law  is  "If  x  41  0,  then  x  =  1", — which  gives 
immediately  "If  #  =  1,  then  x  =  0" — "If  x  is  not  false,  then  x  is  true, 
and  if  x  is  not  true,  then  x  is  false".  Peirce  uses  v  and  /  for  "true"  and 
"false",  instead  of  1  and  0,  and  the  law  is  stated  in  the  form 

(x-f)(v-x}  =  0 

But  the  calculus  of  prepositional  functions,  though  derived  from  the 
algebra  for  propositions,  is  not  identical  with  it.  "a;  is  a  man"  is  neither 
true  nor  false.  A  propositional  function  may  be  true  in  some  cases,  false 
in  some  cases.  "  If  a;  is  a  man,  then  #  is  a  mortal "  is  true  in  all  cases,  or 
true  of  any  x;  "a;  is  a  man"  is  true  in  some  cases,  or  true  for  some  values 
of  x.  For  reasons  already  suggested, 

1ix(px  =  <pXi  +  (f>Xz  +  <px3  +  .  .  . 

2x(px  represents  "  <px  is  true  for  some  value  of  the  variable,  x — that  is, 
either  <px\  is  true  or  <?x2  is  true  or  <f>xs  is  true  or  .  .  . "  Similarly, 

Ux<f>X   =    (fXi  X  <pXz  X  <pX3  X  ... 

nx  <px  represents  "  <px  is  true  for  all  values  of  the  variable,  x — that  is, 
<f>Xi  is  true  and  <px2  is  true  and  <px3  is  true  and  ..." 

If  (a)xy,  or  more  conveniently,   <p(x,  y),  represent  "x  is  agent  of  y", 
and  (b)xy,  or  more  conveniently,  \f/(x,  y),  mean  "x  is  benefactor  of  y",  then 

nx2y[<p(x,  y)  x$(x,  y)] 

will  mean  that  for  all  values  of  x  and  some  values  of  y,  "x  is  agent  of  y 
and  x  is  benefactor  of  y"  is  true — that  is,  it  represents  the  proposition 
"Everyone  is  both  agent  and  benefactor  of  someone".  This  will  appear 
if  we  expand  nx2y[(p(x,  y)  x\f/(x,  y)]: 

nx^y[<p(x,  y}  x^(ar,  y}} 

I,  2/0  X<KZI}  yi)]  +  [<f>(xi,  2/2)  *t(xi,  2/2)]+  ...} 

/i)  x  t(x2,  2/1)]  +  [  <p(x<i,  2/2)  x  t(xz,  2/2)]  +  .  .  . } 
x  {[<f>(x3,  yi)  x^(z3,  yi)]  +  [v(x3,  1/2)  x^(z3,  2/2)]  +  ...} 
x  ...  Etc.,  etc. 

This  expression  reads  directly  "  {Either  [xt  is  agent  of  y\  and  x\  is  bene- 
factor of  2/1]  or  [xi  is  agent  of  2/2  and  x\  is  benefactor  of  2/2]  or  .  .  . }  and  [either 
[#2  is  agent  of  2/1  and  x2  is  benefactor  of  2/1]  or  [x2  is  agent  of  2/2  and  z2  is  bene- 
factor of  2/2]  or  .  .  .}  and  {either  [x3  is  agent  of  2/1  and  x3  is  benefactor  of 
2/i]  or  [xs  is  agent  of  2/2  and  xs  is  benefactor  of  2/2]  or  .  .  . }  and  .  .  .  Etc.,  etc". 


98  A  Survey  of  Symbolic  Logic 

The  operator  S,  which  is  nearer  the  argument,  or  "Boolian"  as  Peirce  calls 
it,  indicates  the  operation,  +  ,  within  the  lines.  The  outside  operator,  II, 
indicates  the  operation,  x,  between  the  lines  —  i.  e.,  in  the  columns;  and 
the  subscript  of  the  operator  nearer  the  Boolian  indicates  the  letter  which 
varies  within  the  lines,  the  subscript  of  the  outside  operator,  the  letter 
which  varies  from  line  to  line.  Three  operators  would  give  a  three-dimen- 
sional array.  With  a  little  patience,  the  reader  may  learn  to  interpret 
any  such  expression  directly  from  the  meaning  of  simple  logical  sums  and 
logical  products.  For  example,  with  the  same  meanings  of  v(x,  y)  and 


will  mean  "Everyone  (x)  is  agent  of  some  (y)  benefactor  of  himself". 
(Note  the  order  of  the  variables  in  the  Boolian.)  And 

2x2vUz[<p(x,  z)  +  iKz,  y)] 

will  symbolize  "There  is  some  x  and  some  y  such  that,  for  every  z,  either 
x  is  agent  of  2  or  2  is  benefactor  of  y"',  or,  more  simply,  "There  is  some 
pair,  x  and  y,  such  that  x  is  agent  of  all  non-benefactors  of  y". 

The  laws  for  the  manipulation  of  such  Boolians  with  n  and  S  operators 
are  given  as  follows  :  149 

"  1st.  The  different  premises  having  been  written  with  distinct  indices 
(the  same  index  not  being  used  in  two  propositions)  are  written  together, 
and  all  the  H's  and  S's  are  to  be  brought  to  the  left.  This  can  evidently  be 
done,  for 


[Or  in.  the  more  convenient,  and  probably  more  familiar,  notation, 
Hx<px  *ny<py  =  Hxny(<f>x  x  <py) 


"2d.  Without  deranging  the  order  of  the  indices  of  any  one  premise, 
the  n's  and  S's  belonging  to  different  premises  may  be  moved  relatively 
to  one  another,  and  as  far  as  possible  the  S's  should  be  carried  to  the  left 

149  Alg.  Log.  1885,  pp.  196-98. 


The  Development  of  Symbolic  Logic  99 

of  the  n's.     We  have 

ILiUjXa  =  UjUiXij  [Or,  UxUy<p(x,  y}  =  UyUx<p(x,  y)] 

WtXij  =  V&iXij  [Or,  -2x-2v<p(x,  y}  =  2y2x<p(x,  y)} 

and  also         2t-n,-;r,-?//  =  H^iXiyj  [Or,  S«ny(^a;x^)  =  H.y2x(<px  x\I/y)] 

But  this  formula  does  not  hold  when  i  and  j  are  not  separated.  We  do 
have,  however, 

SiHiXij  -<  Tlj^iXij        [Or,  SJIy^fo  y)  c.HyVx<p(x,  y)] 

It  will,  therefore,  be  well  to  begin  by  putting  the  2's  to  the  left  as  far  as 
possible,  because  at  a  later  stage  of  the  work  they  can  be  carried  to  the 
right  but  not  [always]  to  the  left.  For  example,  if  the  operators  of  two 
premises  are  IlfSjTIfc  and  2^11^2^,  we  can  unite  them  in  either  of  the  two 
orders 


and   shall  usually  obtain   different  conclusions  accordingly.     There  will 
often  be  room  for  skill  in  choosing  the  most  suitable  arrangement. 

.  .  .  "5th.  The  next  step  consists  in  multiplying  the  whole  Boolian 
part,  by  the  modification  of  itself  produced  by  substituting  for  the  index 
of  any  II  any  other  index  standing  to  the  left  of  it  in  the  Quantifier.  Thus, 
for 

SJI/Zi/  [Or,  for  2xTLv<p(x,  y}, 

we  can  write        2tII,-Z,-,7t-i  SJIy{  <p(x,  y}  x  <p(x,  x)}] 

"6th.  The  next  step  consists  in  the  re-manipulation  of  the  Boolian 
part,  consisting,  1st,  in  adding  to  any  part  any  term  we  like;  2d,  in  dropping 
from  any  part  any  factor  we  like,  and  3d,  in  observing  that 

xx  =  f,        x  +  x  =  v, 
so  that  xxy  +  z  =  z         (x  +  x  +  y}z  =  z 

"7th.  n's  and  2's  in  the  Quantifier  whose  indices  no  longer  appear  in 
the  Boolian  are  dropped. 

"The  fifth  step  will,  in  practice,  be  combined  with  part  of  the  sixth 
and  seventh.  Thus,  from  2tIIj7t-j  we  shall  at  once  proceed  to  2^.-t  if  we  like." 

We  may  say,  in  general,  that  the  procedures  which  are  valid  in  this 
calculus  are  those  wrhich  can  be  performed  by  treating  2x<px  as  a  sum, 
<pxi  +  <pxz  +  <f>x3  +  .  .  .,  and  Ux<px  as  a  product,  <pxi  x  <pxz  x  <px3  x  .  .  .  ; 
I,xny\f/(x,  y)  as  a  sum,  for  the  various  values  of  x,  of  products,  each  for 


100  A  Survey  of  Symbolic  Logic 

the  various  values  of  y,  and  so  on.  Thus  this  calculus  may  be  derived  from 
the  calculus  of  propositions.  But  Peirce  does  not  carry  out  any  proofs 
of  the  principles  of  the  system,  and  he  notes  that  this  method  of  proof 
would  be  theoretically  unsound.150  "It  is  to  be  remarked  that  St£t-  and 
HiXi  are  only  similar  to  a  sum  and  a  product;  they  are  not  strictly  of  that 
nature,  because  the  individuals  of  the  universe  may  be  innumerable." 

Another  way  of  saying  the  same  thing  would  be  this:  The  laws  of  the 
calculus  of  propositions  cannot  extend  to  2iXi  and  nt:rt-,  because  the  extension 
of  these  laws  to  aggregates  in  general,  by  the  method  which  the  mathemati- 
cal analogies  of  sum  and  product  suggest,  would  require  the  principle  of 
mathematical  induction,  which  is  not  sufficient  for  proof  in  case  the  aggre- 
gate is  infinite. 

The  whole  of  the  calculus  of  relatives  may  be  derived  from  this  calculus 
of  prepositional  functions  by  the  methods  which  have  been  exemplified — 
that  is,  by  representing  any  relation,  6,  as  S<Sy(6)<,-(7  :  J),  and  defining  the 
relations,  such  as  "converse  of",  "relative-product,"  etc.,  which  dis- 
tinguish the  calculus,  as  II  and  2  functions  of  the  elementary  relatives. 
We  need  not  enter  into  the  detail  of  this  matter,  since  Sections  II  and  III 
of  Chapter  IV  will  develop  the  calculus  of  propositional  functions  by  a 
modification  of  Peirce's  method,  while  Section  IV  of  that  chapter  will  show 
how  the  calculus  of  classes  can  be  derived  from  this  calculus  of  propositional 
functions,  Section  V  will  indicate  the  manner  in  which  the  calculus  of  rela- 
tions may  be  similarly  derived,  and  Section  VI  will  suggest  how,  by  a 
further  important  modification  of  Peirce's  method,  a  theoretically  adequate 
logic  of  mathematics  may  be  obtained. 

It  remains  to  consider  briefly  Peirce's  studies  toward  the  derivation  of 
other  mathematical  relations,  operations,  and  systems  from  symbolic  logic. 
The  most  important  paper,  in  this  connection,  is  "Upon  the  Logic  of 
Mathematics".151  Certain  portions  of  the  paper,  "On  an  Improvement  in 
Boole's  Calculus  of  Logic",  and  of  the  monograph,  "Description  of  a  Nota- 
tion for  the  Logic  of  Relatives",  are  also  of  interest. 

The  first-mentioned  of  these  is  concerned  to  show  how  the  relations 
+  ,  =,  etc.,  of  arithmetic  can  be  defined  in  terms  of  the  corresponding  logi- 
cal relations,  and  the  properties  of  arithmetical  relations  deduced  from 
theorems  concerning  their  logical  analogues.162 

"Imagine  ...  a  particular  case  under  Boole's  calculus,  in  which  the 

160  Alg.  Log.  1885,  p.  195. 

161  Proc.  Amer.  Acad.,  vii,  402-12. 
152  Loc.  cit.,  pp.  410-11. 


The  Development  of  Symbolic  Logic  101 

letters  are  no  longer  terms  of  first  intention,  but  terms  of  second  intention, 
and  that  of  a  special  kind.  .  .  .  Let  the  letters  .  .  .  relate  exclusively  to 
the  extension  of  first  intensions.  Let  the  differences  of  the  characters  of 
things  and  events  be  disregarded,  and  let  the  letters  signify  only  the  differ- 
ences of  classes  as  wider  or  narrower.  In  other  words,  the  only  logical 
comprehension  which  the  letters  considered  as  terms  will  have  is  the  greater 
or  less  divisibility  of  the  class.  Thus,  n  in  another  case  of  Boole's  calculus 
might,  for  example,  denote  'New  England  States';  but  in  the  case  now 
supposed,  all  the  characters  which  make  these  states  what  they  are,  being 
neglected,  it  would  signify  only  what  essentially  belongs  to  a  class  which 
has  the  same  relation  to  higher  and  lower  classes  which  the  class  of  New 
England  States  has, — that  is,  a  collection  of  six. 

"In  this  case,  the  sign  of  identity  will  receive  a  special  meaning.  For, 
if  m  denotes  what  essentially  belongs  to  a  class  of  the  rank  of  'sides  of  a 
cube',  then  [the  logical]  m  =  n  will  imply,  not  that  every  New  England 
State  is  the  side  of  a  cube,  and  conversely,  but  that  whatever  essentially 
belongs  to  a  class  of  the  numerical  rank  of  '  New  England  States '  essentially 
belongs  to  a  class  of  the  rank  of  '  sides  of  a  cube ',  and  conversely.  Identity 
of  this  particular  sort  may  be  termed  equality.  .  .  . " 

If  a,  b,  c,  etc.  represent  thus  the  number  of  the  classes,  a,  b,  c,  etc., 
then  the  arithmetical  relations  can  be  defined  as  logical  relations.  The 
logical  relation  a +  6,  already  defined,  will  represent  arithmetical  addition: 
And  from  the  fact  that  the  logical  +  is  commutative  and  associative,  it 
will  follow  that  the  arithmetical  +  is  so  also.  Arithmetical  multiplication 
is  more  difficult  to  deal  with  but  may  be  defined  as  follows : 153 

a  X  b  represents  an  event  when  a  and  b  are  events  only  if  these  events 
are  independent  of  each  other,  in  which  case  a  X  b  =  a  b  [where  a  6  is  the 
logical  product].  By  the  events  being  independent  is  meant  that  it  is 
possible  to  take  two  series  of  terms,  A\,  Az,  As,  etc.,  and  B\,  B2,  B3,  etc., 
such  that  the  following  conditions  are  satisfied.  (Here  x  denotes  any 
individual  or  class,  not  nothing;  Am,  An>  Bm,  Bn,  any  members  of  the 
two  series  of  terms,  and  2  A,  2  B,  S  (A  B)  logical  sums  of  some  of  the 
An's,  the  Bn's,  and  the  (An  5»)'s  respectively.) 

Condition  1.  No  Am  is  An 

2.  No  Bm  is  Bn 

3.  x  =  S  (A  B) 

4.  a  =  S  A 
163  Loc.  tit.,  p.  403. 


102  A  Survey  of  Symbolic  Logic 

Condition  5.     b  =  2  B 

6.     Some  Am  is  Bn 

This  definition  is  somewhat  involved:  the  crux  of  the  matter  is  that 
a  b  will,  in  the  case  described,  have  as  many  members  as  there  are  combina- 
tions of  a  member  of  a  with  a  member  of  b.  Where  the  members  of  a  are 
distinct  (condition  1)  and  the  members  of  b  are  distinct  (condition  2),  these 
combinations  will  be  of  the  same  multitude  as  the  arithmetical  a  X  b. 

It  is  worthy  of  remark  that,  in  respect  both  to  addition  and  to  multi- 
plication, Peirce  has  here  hit  upon  the  same  fundamental  ideas  by  means 
of  which  arithmetical  relations  are  defined  in  Principia  Mathematical 
The  "second  intention"  of  a  class  term  is,  in  Principia,  Nc'a;  a  +  b,  in 
Peirce's  discussion,  corresponds  to  what  is  there  called  the  "arithmetical 
sum"  of  two  logical  classes,  and  a  X  &  to  what  is  called  the  "arithmetical 
product".  But  Peirce's  discussion  does  not  meet  all  the  difficulties — that 
could  hardly  be  expected  in  a  short  paper.  In  particular,  it  does  not 
define  the  arithmetical  sum  in  case  the  classes  summed  have  members  in 
common,  and  it  does  not  indicate  the  manner  of  defining  the  number  of  a 
class,  though  it  does  suggest  exactly  the  mode  of  attack  adopted  in  Prin- 
cipia, namely,  that  number  be  considered  as  a  property  of  cardinally  similar 
classes  taken  in  extension. 

The  method  suggested  for  the  derivation  of  the  laws  of  various  numerical 
algebras  from  those  of  the  logic  of  relatives  is  more  comprehensive,  though 
here  it  is  only  the  order  of  the  systems  which  is  derived  from  the  order  of 
the  logic  of  relatives;  there  is  no  attempt  to  define  the  number  or  multitude 
of  a  class  in  terms  of  logical  relations.155 

We  are  here  to  take  a  closed  system  of  elementary  relatives,  every 
individual  in  which  is  either  a  T  or  a  P  and  none  is  both. 

Let  c  =  (T  :  T) 
8  =  (P:P) 

p  =  (P:T) 
t  =  (T  :P) 

Suppose  T  here  represent  an  individual  teacher,  and  P  an  individual  pupil: 
the  system  will  then  be  comparable  to  a  school  in  which  every  person  is 
either  teacher  or  pupil,  and  none  is  both  and  every  teacher  teaches  every 
pupil.  The  relative  term,  c,  will  then  be  defined  as  the  relation  of  one 

154  See  Vol.  n,  Section  A. 

155  "  Description  of  a  Notation,  pp.  359  ff. 


The  Development  of  Symbolic  Logic  103 

teacher  to  another,  that  is,  "colleague".  Similarly,  s  is  (P  :  P),  the  rela- 
tion of  one  pupil  to  another,  that  is,  "schoolmate".  The  relative  term,  p, 
is  (P  :  T),  the  relation  of  any  pupil  to  any  teacher,  that  is,  "pupil".  And 
the  relative  term,  t,  is  (T  :  P),  the  relation  of  any  teacher  to  any  pupil, 
that  is,  "teacher".  Thus  from  the  two  non-relative  terms,  T  and  P,  are 
generated  the  four  elementary  relatives,  c,  s,  t,  and  p. 

The  properties  of  this  system  will  be  clearer  if  we  venture  upon  certain 
explanations  of  the  properties  of  elementary  relatives — which  Peirce  does 
not  give  and  to  the  form  of  which  he  might  object.  For  any  such  relative 
(I  :  J),  where  the  7's  and  the  /'s  are  distinct,  we  shall  have  three  laws: 

(1)  (7  :  J)  J  =  I 

Whatever  has  the  (7  :  J)  relation  to  a  J  must  be  an  7:  whoever  has  the 
teacher-pupil  relation  to  a  pupil  must  be  a  teacher. 

(2)  (7  :  J)  7  =  0 

Whatever  has  the  teacher-pupil  relation  to  a  teacher  (where  teachers  and 
pupils  are  distinct)  does  not  exist. 

(3)  (7:J)   (#:7<0  =  ((I  :  J)\H]  :  K 

The  relation  of  those  which  have  the  (7  :  J)  relation  to  those  which  have 
the  (H  :  K)  relation  is  the  relation  of  those-which-have-the-(7  :  J)-relation- 
to-an-77  to  a  K. 

It  is  this  third  law  which  is  the  source  of  the  important  properties  of 
the  system.  For  example: 

t\p  =  (T  :P)|(P  :  T)  =  [(T  :  P)  |P]  :  T  =  (T  :  T)  =  c 

The  teachers  of  any  person's  pupils  are  that  person's  colleagues.  (Our 
illustration,  to  fit  the  system,  requires  that  one  may  be  his  own  colleague 
or  his  own  schoolmate.) 

c  c  =  (T  :  T)  |  (T  :  T)  =  [(T  :  T)  \  T]  :  T  =  (T  :  T)  =  c 
The  colleagues  of  one's  colleagues  are  one's  colleagues. 

t\t  =  (T  :P)\(T  :  P)  =  [(T  :  P)  T]  :  P  =  (0  :  P)  =  0 
There  are  no  teachers  of  teachers  in  the  system. 

p  s  =  (P  :  T)  (P  :  P)  =  [(P  :  T)  |P]  :  P  =  (0  :  P)  =  0 

There  are  no  pupils  of  anyone's  schoolmates  in  the  system. 

The  results  may  be  summarized  in  the  following  multiplication  table, 
in  which  the  multipliers  are  in  the  column  at  the  right  and  the  multiplicands 


104  A  Survey  of  Symbolic  Logic 

at  the  top  (relative  multiplication  not  being  commutative) : 156 

t     p    s 


c     t     0     0 
0     0     c     t 


p    s     0    0 
0    0    p    s 

The  symmetry  of  the  table  should  be  noted.  The  reader  may  easily  in- 
terpret the  sixteen  propositions  which  it  gives. 

To  the  algebra  thus  constituted  may  be  added  modifiers  of  the  terms, 
symbolized  by  small  roman  letters.  If  f  is  "French",  f  will  be  a  modifier 
of  the  system  in  case  French  teachers  have  only  French  pupils,  and  vice 
versa.  Such  modifiers  are  "scalars"  of  the  system,  and  any  expression  of 
the  form 

a  c  +  b  t+  c  p  +  d  s 

where  c,  t,  p,  and  s  are  the  relatives,  as  above,  and  a,  b,  c,  d  are  scalars, 
Peirce  calls  a  "logical  quaternion".  The  product  of  a  scalar  with  a  term 
is  commutative, 

bt  =  tb 

since  this  relation  is  that  of  the  non-relative  logical  product.  Inasmuch  as 
any  (dyadic,  triadic,  etc.)  relative  is  resolvable  into  a  logical  sum  of  (pairs, 
triads,  etc.)  elementary  relatives,  it  is  plain  that  any  general  relative  what- 
ever is  resolvable  into  a  sum  of  logical  quaternions. 

If  we  consider  a  system  of  relatives,  each  of  which  is  of  the  form 

ai  +  bj  +  ck  +  dl+  ... 
where  i,  j,  k,  I,  etc.  are  each  of  the  form 

mu  +  nv  +  o  w+  ... 

where  m,  n,  o,  etc.  are  scalars,  and  u,  v,  w,  etc.  are  elementary  relatives, 
we  shall  have  a  more  complex  algebra.  By  such  processes  of  complication, 
multiple  algebras  of  various  types  can  be  generated.  In  fact,  Peirce  says: 157 
"I  can  assert,  upon  reasonable  inductive  evidence,  that  all  such  [linear 
associative]  algebras  can  be  interpreted  on  the  principles  of  the  present 
notation  in  the  same  way  as  those  given  above.  In  other  words,  all  such 
algebras  are  complications  and  modifications  of  the  algebra  of  (156)  [for 
which  the  multiplication  table  has  been  given].  It  is  very  likely  that  this 

™Ibid.,  p.  361. 

167  Ibid.,  pp.  363-64. 


The  Development  of  Symbolic  Logic  105 

is  true  of  all  algebras  whatever.  The  algebra  of  (156),  which  is  of  such  a 
fundamental  character  in  reference  to  pure  algebra  and  our  logical  nota- 
tion, has  been  shown  by  Professor  [Benjamin]  Peirce  to  be  the  algebra  of 
Hamilton's  quaternions." 

Peirce  gives  the  form  of  the  four  fundamental  factors  of  quaternions  and 
of  scalars,  tensors,  vectors,  etc.,  with  their  logical  interpretations  as  relative 
terms  with  modifiers  such  as  were  described  above. 

One  more  item  of  importance  is  Peirce's  modification  of  Boole's  calculus 
of  probabilities.  This  is  set  forth  with  extreme  brevity  in  the  paper,  "  On 
an  Improvement  in  Boole's  Calculus  of  Logic".158  For  the  expression  of 
the  relations  involved,  we  shall  need  to  distinguish  the  logical  relation  of 
identity  of  two  classes  in  extension  from  the  relation  of  numerical  equality. 
We  may,  then,  express  the  fact  that  the  class  a  has  the  same  membership 
as  the  class  b,  or  all  a's  are  all  6's,  by  a  =•  6,  and  the  fact  that  the  number 
of  members  of  a  is  the  same  as  the  number  of  members  of  b,  by  a  =  b. 
Also  we  must  remember  the  distinction  between  the  logical  relations  ex- 
pressed by  a  +  b,  ab,  a  \-b,  and  the  corresponding  arithmetical  relations 
expressed  by  a  +  b,  a  X  b,  and  a  —  b.  Peirce  says: 169 

"Let  every  expression  for  a  class  have  a  second  meaning,  which  is  its 
meaning  in  a  [numerical]  equation.  ^Namely,  let  it  denote  the  proportion 
of  individuals  of  that  class  to  be  found  among  all  the  individuals  examined 
in  the  long  run. 

"Then  we  have 

If  a  =  b        a  =  b 

a-{-b  =  (a  +  b)-\-ab 

"Let  ba  denote  the  frequency  of  the  6's  among  the  a's.  Then  considered 
as  a  class,  if  a  and  b  are  events  ba  denotes  the  fact  that  if  a  happens  b  happens. 

a  X  ba  =  a  b 

"  It  will  be  convenient  to  set  down  some  obvious  and  fundamental  proper- 
ties of  the  function  ba. 

a  X  ba  =  6  X  ab 
<p(ba,  ca)  =  <p(b,  c)a 
(1  -  6).  =  1  -  ba 

ba  =  -  +  fcj-c  (  1  -  - 
a  V         a 

IM  Proc.  Amer.  Acad.,  vn,  255  ff. 
359  Ibid.,  pp.  255-56. 


106  A  Survey  of  Symbolic  Logic 


1  -  a 
ab  =  1  -    —7  —  X  0(i 


).=  Ml))." 

The  chief  points  of  difference  between  this  modified  calculus  of  prob- 
abilities and  the  original  calculus  of  Boole  are  as  follows: 

(1)  Where  Boole  puts  p,  q,  etc.  for  the  "probability  of  a,  of  b,  etc.", 
in  passing  from  the  logical  to  the  arithmetical  interpretation  of  his  equa- 
tions, Peirce  simply  changes  the  relations  involved  from  logical  relations  to 
the  corresponding  arithmetical  relations,  in  accordance  with  the  foregoing, 
and  lets  the  terms  a,  b,  etc.  stand  for  the  frequency  of  the  a's,  b's,  etc. 
in  the  system  under  discussion. 

(2)  Boole  has  no  symbol  for  the  frequency  of  the  a's  amongst  the  6's, 
which  Peirce  represents  by  ab.     As   a   result,  Boole   is  led   to   treat   the 
probabilities  of  all  unconditioned  simple  events  as  independent  —  a  pro- 
cedure which  involved  him  in  many  difficulties  and  some  errors. 

(3)  Peirce  has  a  complete  set  of    four    logical  operations,  and    four 
analogous  operations  of  arithmetic.     This  greatly  facilitates  the  passage 
from  the  purely  logical  expression  of  relations  of  classes  or  events  to  the 
arithmetical  expression  of  their  relative  frequencies  or  probabilities. 

Probably  there  is  no  one  piece  of  work  which  would  so  immediately 
reward  an  investigator  in  symbolic  logic  as  would  the  development  of  this 
calculus  of  probabilities  in  such  shape  as  to  make  it  simple  and  practicable. 
Except  for  a  monograph  by  Poretsky  and  the  studies  of  H.  MacColl,160 
the  subject  has  lain  almost  untouched  since  Peirce  wrote  the  above  in  1867. 

Peirce's  contribution  to  our  subject  is  the  most  considerable  of  any  up 
to  his  time,  with  the  doubtful  exception  of  Boole's.  His  papers,  however, 
are  brief  to  the  point  of  obscurity:  results  are  given  summarily  with  little 
or  no  explanation  and  only  infrequent  demonstrations.  As  a  consequence, 
the  most  valuable  of  them  make  tremendously  tough  reading,  and  they 
have  never  received  one-tenth  the  attention  which  their  importance  de- 
serves.161 If  Peirce  had  been  given  to  the  pleasantly  discursive  style  of 
De  Morgan,  or  the  detailed  and  clearly  accurate  manner  of  Schroder,  his 
work  on  symbolic  logic  would  fill  several  volumes. 

160  Since  the  above  was  written,  a  paper  by  Couturat,  posthumously  published,  gives 
an  unusually  clear  presentation  of  the  fundamental  laws  of  probability  in  terms  of  symbolic 
logic.     See  Bibl. 

161  Any  who  find  our  report  of  Peirce's  work  unduly  difficult  or  obscure  are  earnestly 
requested  to  consult  the  original  papers. 


The  Development  of  Symbolic  Logic  107 

VIII.     DEVELOPMENTS  SINCE  PEIRCE 

Contributions  to  symbolic  logic  which  have  been  made  since  the  time 
of  Peirce  need  be  mentioned  only  briefly.  These  are  all  accessible  and  in  a 
form  sufficiently  close  to  current  notation  to  be  readily  intelligible.  Also, 
they  have  not  been  superseded,  as  have  most  of  the  papers  so  far  discussed ; 
consequently  they  are  worth  studying  quite  apart  from  any  relation  to 
later  work.  And  finally,  much  of  the  content  and  method  of  the  most 
important  of  them  is  substantially  the  same  with  what  will  be  set  forth  in 
later  chapters,  or  is  such  that  its  connection  with  what  is  there  set  forth 
will  be  pointed  out.  But  for  the  sake  of  continuity  and  perspective,  a 
summary  account  may  be  given  of  these  recent  developments. 

We  should  first  mention  three  important  pieces  of  work  contemporary 
with  Peirce's  later  treatises.162 

Robert  Grassmann  had  included  in  his  encyclopedic  Wissenschaftslehre 
a  book  entitled  Die  Begriffslehre  oder  Logik,163  containing  (1)  Lehre  von  den 
Begriffen,  (2)  Lehre  von  den  Urtheilen,  and  (3)  Lehre  von  den  Schlussen. 
The  Begriffslehre  is  the  second  book  of  Die  Formenlehre  oder  Mathematik, 
and  as  this  would  indicate,  the  development  of  logic  is  entirely  mathematical. 
An  important  character  of  Grassmann's  procedure  is  the  derivation  of  the 
laws  of  classes,  or  Begriffe,  as  he  insists  upon  calling  them,  from  the  laws 
governing  individuals.  For  example,  the  laws  a  +  a  =  a  and  a- a  =  a, 
where  a  is  a  class,  are  derived  from  the  laws  e  +  e  =  e,  e-e  =  e,  e^e-i  =  0, 
where  e,  e\,  e^  represent  individuals.  This  method  has  much  to  commend 
it,  but  it  has  one  serious  defect — the  supposition  that  a  class  can  be  treated 
as  an  aggregate  of  individuals  and  the  laws  of  such  aggregates  proved 
generally  by  mathematical  induction.  As  Peirce  has  observed,  this  method 
breaks  down  when  the  number  of  individuals  may  be  infinite.  Another 
difference  between  Grassmann  and  others  is  the  use  throughout  of  the 
language  of  intension.  But  the  method  and  the  laws  are  those  of  extension, 
and  in  the  later  treatise,  there  are  diagrammatic  illustrations  in  which 
"concepts"  are  represented  by  areas.  Although  somewhat  incomplete,  in 

162  Alexander  MacFarlane,  Principles  of  the  Algebra  of  Logic,  1879,  gives  a  masterly 
presentation  of  the  Boolean  algebra.     There  are  some  notable  extensions  of  Boole's  methods 
and  one  or  two  emendations,  but  in  general  it  is  the  calculus  of  Boole  unchanged.     Mac- 
Farlane's  paper  "On  a  Calculus  of  Relationship"  (Proc.  Roy.  Soc.  Edin.,  x,  224-32)  re- 
sembles somewhat,  in  its  method,  Peirce's  treatment  of  "elementary  relatives".     But 
the  development  of  it  seems  never  to  have  been  continued. 

163  There  are  two  editions,  1872  and  1890.     The  later  is  much  expanded,  but  the  plan 
and  general  character  is  the  same. 


108  A  Survey  of  Symbolic  Logic 

other  respects  Grassmann's  calculus  is  not  notably  different  from  others 
which  follow  the  Boolean  tradition. 

Hugh  MacColl's  first  two  papers  on  "The  Calculus  of  Equivalent 
Statements",164  and  his  first  paper  "On  Symbolical  Reasoning",165  printed 
in  1878-80,  •  present  a  calculus  of  propositions  which  has  essentially  the 
properties  of  Peirce's,  without  II  and  S  operators.  In  others  words,  it  is 
a  calculus  of  propositions,  like  the  Two-Valued  Algebra  of  Logic  as  we  know 
it  today.  And  the  date  of  these  papers  indicates  that  their  content  was 
arrived  at  independently  of  Peirce's  studies  which  deal  with  this  tonic. 
In  fact,  MacColl  writes,  in  1878,  that  he  has  not  seen  Boole.166 

The  calculus  set  forth  in  MacColl's  book,  Symbolic  Logic  and  its 
Applications,16"7  is  of  an  entirely  different  character.  Here  the  funda- 
mental symbols  represent  prepositional  functions  rather  than  propositions; 
and  instead  of  the  two  traditional  truth  values,  "true"  and  "false",  we 
have  "true",  "false",  "certain",  "impossible"  and  "variable"  (not  cer- 
tain and  not  impossible).  These  are  indicated  by  the  exponents  r,  i,  f, 
17,  6  respectively.  The  result  is  a  highly  complex  system,  the  fundamental 
ideas  and  procedures  of  which  suggest  somewhat  the  system  of  Strict 
Implication  to  be  set  forth  in  Chapter  V. 

The  calculus  of  Mrs.  Ladd-Franklin,  set  forth  in  the  paper  "On  the 
Algebra  of  Logic"  in  the  Johns  Hopkins  studies,168  differs  from  the  other 
systems  based  on  Boole  by  the  use  of  the  copula  v .  Where  a  and  b  are 
classes,  a  v6  represents  "a  is-partly  6",  or  "Some  a  is  6",  and  its  negative, 
a  v  b,  represents  "  a  is-wholly-not-6  ",  or  "  No  a  is  b  ".  Thus  a  v  b  is  equiva- 
lent to  a  b  3=  0,  and  a  v  b  to  a  b  =  0.  These  two  relations  can,  between 
them,  express  any  assertable  relation  in  the  algebra,  a  c  b  will  be  a  v  -b, 
and  a  =  b  is  represented  by  the  pair,  (a  v-6)(-a  v6).  For  propositions, 
a  v  6  denotes  that  a  and  b  are  consistent — a  does  not  imply  that  b  is  false 
and  b  does  not  imply  that  a  is  false.  And  avb  symbolizes  "a  and  b  are 
inconsistent" — if  a  is  true,  b  is  false;  if  b  is  true,  a  is  false.  The  use  of  the 
terms  "consistent"  and  "inconsistent"  in  this  connection  is  possibly  mis- 
leading: any  two  true  propositions  or  any  two  false  propositions  are  con- 

1M  (1)  Proc.  London  Math.  Soc.,  ix,  9-20;   (2)  ibid.,  ix,  177-86. 

168  Mind,  v  (1880),  45-60. 

168  Proc.  London  Math.  Soc.,  rx,  178. 

147  Longmans,  1906. 

168  The  same  volume  contains  an  interesting  and  somewhat  complicated  system  by 
O.  H.  Mitchell.  Peirce  acknowledged  this  paper  as  having  shown  us  how  to  express  uni- 
versal and  particular  propositions  as  II  and  S  functions.  B.  I.  Oilman's  study  of  relative 
number,  also  in  that  volume,  belongs  to  the  number  of  those  papers  which  are  important 
in  connecting  symbolic  logic  with  the  theory  of  probabilities. 


The  Development  of  Symbolic  Logic  109 

sistent  in  this  sense,  and  any  two  propositions  one  of  which  is  true  and 
the  other  false  are  inconsistent.  This  is  not  quite  the  usual  meaning  of 
"consistent"  and  "inconsistent" — it  is  related  to  what  is  usually  meant  by 
these  terms  exactly  as  the  "material  implication  acb  is  related  to  what 
is  usually  meant  by  "6  can  be  inferred  from  a". 

That  a  given  class,  x,  is  empty,  or  a  given  proposition,  x,  is  false,  x  =  0, 
may  be  expressed  by  x  v  oo,  where  °o  is  "everything" — in  most  systems 
represented  by  1.  That  a  class,  y,  has  members,  is  symbolized  by  yv  oo. 
This  last  is  of  doubtful  interpretation  where  y  is  a  proposition,  since  Mrs. 
Ladd-Franklin's  system  does  not  contain  the  assumption  which  is  true 
for  propositions  but  not  for  classes,  usually  expressed,  "If  x  =(=  0,  then 
x  =  1,  and  if  x  4=  1>  then  x  =  0".  x  v  oo  may  be  abbreviated  to  x  v, 
a  6  v  oo  to  a  6  v ,  and  ?/  v  <x>  to  y  v ,  c  dv  <x>  to  c  dv ,  etc.,  since  it  is  always 
understood  that  if  one  term  of  a  relation  v  or  v  is  missing,  the  missing 
term  is  °o .  This  convention  leads  to  a  very  pretty  and  convenient  opera- 
tion: v  or  v  may  be  moved  past  its  terms  in  either  direction.  Thus, 

(a  v6)  =  (ab  v)  =  (  va  6) 
and         (%vy)  =  (xyv)  =  (  v  x  y) 

But  the  forms  (  v  a  b)  and  (  v  x  y)  are  never  used,  being  redundant  both 
logically  and  psychologically. 

Mrs.  Ladd-Franklin's  system  symbolizes  the  relations  of  the  traditional 
logic  particularly  well : 

All  a  is  b.  a  v  -b,  or  a  -b  v 

No  a  is  b.  avb,  or  a  6  v 

Some  a  is  b.  avb,  or  a  b  v 

Some  a  is  not  b.  a  v  -b,  or  a  -6  v 

Thus  v  characterizes  a  universal,  v  a  particular  proposition.  And  any 
pair  of  contradictories  will  differ  from  one  another  simply  by  the  difference 
between  v  and  v .  The  syllogism,  "  If  all  a  is  6  and  all  b  is  c,  then  all 
a  is  c, "  will  be  represented  by 

(a  v  -b)  (b  v  -c)  v  (a  v  c) 

where  v ,  or  v ,  within  the  parentheses  is  interpreted  for  classes,  and  v 
between  the  parentheses  takes  the  prepositional  interpretation.  This  ex- 
pression may  also  be  read,  "'All  a  is  b  and  all  b  is  c'  is  inconsistent  with 
the  negative  (contradictory)  of  'Some  a  is  not  c'".  It  is  equivalent  to 

(a  v  -b)  (b  v  -c)  (a  v  -c)  v 


110  A  Survey  of  Symbolic  Logic 

"The  three  propositions,  'All  a  is  &',  'All  b  is  c, '  and  'Some  a  is  not  c', 
are  inconsistent — they  cannot  all  three  be  true".  This  expresses  at  once 
three  syllogisms: 

(1)  (a  v  -b)  (b  v  -c)  v  (a  v  -c) 

"If  all  a  is  b  and  all  6  is  c,  then  all  a  is  c"; 

(2)  (a  v  -6)  (a  v  -c)  v  (b  v  -c) 

" If  all  a  is  b  and  some  a  is  not  c,  then  some  b  is  not  c"; 

(3)  (6  v  -c)  (a  v  -c)  v  (a  v  -6) 

"  If  all  6  is  c  and  some  a  is  not  c,  then  some  a  is  not  b  ". 

Also,  this  method  gives  a  perfectly  general  formula  for  the  syllogism 

(a  v  -b)  (b  v  c)  (a  v  c)  v 

where  the  order  of  the  parentheses,  and  their  position  relative  to  the  sign  v 
which  stands  outside  the  parentheses,  may  be  altered  at  will.  This  single 
rule  covers  all  the  .modes  and  figures  of  the  syllogism,  except  the  illicit 
particular  conclusion  drawn  from  universal  premises.  We  shall  revert  to 
this  matter  in  Chapter  III.169 

The  copulas  v  and  v  have  several  advantages  over  their  equivalents, 
=  0  and  =t=  0,  or  c  and  its  negative:  (1)  v  and  v  are  symmetrical  rela- 
tions whose  terms  can  always  be  interchanged;  (2)  the  operation,  mentioned 
above,  of  moving  v  and  v  with  respect  to  their  terms,  accomplishes  trans- 
formations which  are  less  simply  performed  with  other  modes  of  expressing 
the  copula;  (3)  for  various  reasons,  it  is  psychologically  simpler  and  more 
natural  to  think  of  logical  relations  in  terms  of  v  and  v  than  in  terms 
of  =  0  and  =|=  0.  But  v  and  v  have  one  disadvantage  as  against  = ,  4=  > 
and  c , — they  do  not  so  readily  suggest  their  mathematical  analogues  in 
other  algebras.  For  better  or  for  worse,  symbolic  logicians  have  not 
generally  adopted  v  and  v . 

Of  the  major  contributions  since  Peirce,  the  first  is  that  of  Ernst  Schroder. 
In  his  Operationskreis  des  Logikkalkuls  (1877),  Schroder  pointed  out  that 
the  logical  relations  expressed  in  Boole's  calculus  by  subtraction  and  divi- 
sion were  all  otherwise  expressible,  as  Peirce  had  already  noted.  The 
meaning  of  +  given  by  Boole  is  abandoned  in  favor  of  that  which  it  now 
has,  first  introduced  by  Jevons.  And  the  "law  of  duality",  which  con- 
nects theorems  which  involve  the  relation  + ,  or  +  and  1,  with  corresponding 
theorems  in  terms  of  the  logical  product  x,  or  x  and  0,  is  emphasized. 

»» See  below,  pp.  188  ff. 


The  Development  of  Symbolic  Logic  111 

(This  parallelism  of  formulae  had  been  noted  by  Peirce,  in  his  first  paper, 
but  not  emphasized  or  made  use  of.) 

The  resulting  system  is  the  algebra  of  logic  as  we  know  it  today.  This 
system  is  perfected  and  elaborated  in  Vorlesungen  iiber  die  Algebra  der 
Logik  (1890-95).  Volume  I  of  this  work  covers  the  algebra  of  classes; 
Volume  II  the  algebra  of  propositions;  and  Volume  III  is  devoted  to  the 
calculus  of  relations. 

The  algebra  of  classes,  or  as  we  shall  call  it,  the  Boole-Schroder  algebra, 
is  the  system  developed  in  the  next  chapter.170  We  have  somewhat  elabo- 
rated the  theory  of  functions,  but  in  all  essential  respects,  we  give  the  algebra 
as  it  appears  in  Schroder.  There  are  two  differences  of  some  importance 
between  Schroder's  procedure  and  the  one  we  have  adopted.  Schroder's 
assumptions  are  in  terms  of  the  relation  of  subsumption,  c ,  instead  of  the 
relations  of  logical  product  and  = ,  which  appear  in  our  postulates.  And, 
second,  Schroder  gives  and  discusses  the  various  methods  of  his  predecessors, 
as  well  as  those  characteristically  his  own. 

The  calculus  of  propositions  (Aussagenkalkul)  is  the  extension  of  the 
Boole-Schroder  algebra  to  propositions  by  a  method  which  differs  little 
from  that  adopted  in  Chapter  IV,  Section  I,  of  this  book. 

The  discussion  of  relations  is  based  upon  the  work  of  Peirce.  But 
Peirce's  methods  are  much  more  precisely  formulated  by  Schroder,  and 
the  scope  of  the  calculus  is  much  extended.  We  summarize  the  funda- 
mental propositions  which  Schroder  gives  for  the  sake  of  comparison  both 
with  Peirce  and  with  the  procedure  we  shall  adopt  in  Sections  II  and  III 
of  Chapter  IV. 

1)  A,  B,  C,  D,  E  ...  symbolize  "elements"  or  individuals.171     These 
are  distinct  from  one  another  and  from  0. 

2)  I1  =  A+B  +  C  +  D+  ... 

I1  symbolizes  the  universe  of  individuals  or  the  universe  of  discourse  of 
the  first  order. 

3)  i,  j,  k,  I,  m,  n,  p,  q  represent  any  one  of  the  elements  A,  B,  C,  D,  ... 
of  I1. 

4)  I1  =  Zii 

i 

170  For  an  excellent  summary  by  Schroder,  see  Abriss  der  Algebra  der  Logik ;  ed.  Dr. 
Eugen  M  tiller,  1909-10.     Parts  i  and  n,  covering  Vols.  i  and  n  of  Schroder's  Vorlesungen, 
have  so  far  appeared. 

171  The  propositions  here  noted  will  be  found  in  Vorlesungen  uber  die  Algebra  der  Logik, 
m,  3-42.     Many  others,  and  much  discussion  of  theory,  have  been  omitted. 


112  A  Survey  of  Symbolic  Logic 

5)  i  :  j  represents  any  two  elements,  i  and  j,  of  I1  in  a  determined  order. 

6)  (i  =  ;)  =  (i:j  =j  :  i),  (i  *  j)  =  (i  •  j  4=  j  •  i) 

for  every  i  and  j. 

7)  i:j*0 

Pairs  of  elements  of  I1  may  be  arranged  in  a  "block": 
A  :  A,  A  :  B,  A  :  C,  A  :  D,  ... 

B  :  A,  B  :  B,  B  :  C,  B  :  D, 
8) 

C  :  A,   C  :B,   C  :  C,   C  :  D,    ... 

D  :  A,  D  :  B,  D  :  C,  D  :  D,   ... 

These  are  the  "individual  binary  relatives". 

I2  =     (A:A)  +  (A:B)  +  (A:C)  +  ... 

+  (B  :  A}  +  (B  :  B}  +  (B  :  C)  +  . 

9) 

+  (C  :  A)  +  (C  :  B)  +  (C  :  C)  +  . . . 

+ 

I2  represents  the  universe  of  binary  relatives. 

10)   I2  =  2,2,  (i  :  j)  =  2<2y  (i  :  j)  =  2,,  (i  :  j) 
9)  and  10)  may  be  summarized  in  a  simpler  notation: 

1  =  ?..{  :  j  =     A  :  A  +  A  :  B  +  A  :  C  +  . . . 

+  B :A  +  B :B  +  B :C+  . 

11) 

+  C  :A  +  C  :B  +  C  :C+  ... 

+ 

12)  i  :j  :  h  will  symbolize  an  "individual  ternary  relative". 

13)  I3    =  2*2/2,  (i  :  j  :  h)  =  2^  i  :  j  :  h 
Various  types  of  ternary  relatives  are 

14)  A  :  A  :  A,     B  :  A  :  A,     A  :  B  :  A,     A  :  A  :  B,    A  :  B  :  C 

It  is  obvious  that  we  may  similarly  define  individual  relatives  of  the 
fourth,  fifth,  ...  or  any  thinkable  order. 


The  Development  of  Symbolic  Logic  113 

The  general  form  of  a  binary  relative,  a,  is 

a  =  Si/  aif  (i  :  j) 

where  at-/  is  a  coefficient  whose  value  is  1  for  those  (i  :  j)  pairs  in  which  i  has 
the  relation  a  to  j,  and  is  otherwise  0. 

1  =  Si/  i  :  j 

0  =  the  null  class  of  individual  binary  relatives. 


(a  6)t-/  =  a,-/  &»•/  (a  +  &),-/  =  a»,-  +  6t-/ 

-a.-/  =  (-a),-/  =  -(a,/) 
(a  1  6)i/  =  2^  aih  bhi  (a  1  &)»•/  =  Uh  (aih  +  &*/) 

The  general  laws  which  govern  prepositional  functions,  or  Aussagen- 
schemata,  such*  as  (ab)a,  SA  aih  bhj,  n*  (a^  +  &&/),  H0  at-/,  Sa  a,-/,  etc.,  are  as 
follows  : 

ylu  symbolizes  any  statement  about  u;  UUAU  will  have  the  value  1  in 
case,  and  only  in  case,  Au  =  1  for  every  u;  2UAU  will  have  the  value  1  if 
there  is  at  least  one  u  such  that  Au  =  1.  That  is  to  say,  IIM^4M  means 
"  Au  for  every  u",  and  2UAU  means  "  Au  for  some  u". 

a)  nuAu  cAvc  2UAU,  -[?,UAU]  c  -Av  c  -fnu^4u] 

p)    HUAU  ^  AvLlujrLUi  2suAu  ^  Ay  T  -*  n..'\  u. 

(The  subscript  u,  in  a  and  /3,  represents  any  value  of  the  variable  u.} 

T)  -[nuAu]  =  su  -4U,  -[s«^tt]  =  nM  -Au 

5)  If  ^4W  is  independent  of  u,  then  IIU^4U  =  A,  and  Su^lu  =  A. 

e)  nu(^  c5«)  =  (A  cnu5tt),  n«(Xu  cfi)  =  (su^tt  c£) 


r?)  Stt(^lMc5)  =  (nu^uc5),  2M(^c5u)  =  UcSM5u) 

0)  Sttfl,  or  S«S,(-4uc5w)  =  (ntt^ttcS,Be) 

u(^u  =  i)  =  (nM^u  =  i),         nu(Au  =  0)  =  (su^M  =  0) 


0 

1    U(AU  =  0)  =  (nu^iu  =  0),         ?U(AU  =  i)  =  (stt^u  =  i) 

172  We  write  I  where  Schroder  has  1';    N  where  he  has  0';    (a  |  6)  for  (a;  6);  (a  f  &) 
for  (a  j  b);  -a  for  a;  ~a  for  a. 

9 


1  14  A  Survey  of  Symbolic  Logic 

\(nuAucUuBu-)} 
K)  Hu(AucBu)c-\  ^  c  SuU,  c  #„) 


(The  reader  should  note  that  IIu(^4uCjBu)  is  "formal  implication",  —  in 
Principia  Mathematica,  (x).(px  3^#.) 

X)  A  SU5U  =  Su  A  Bu,  A  +  UUBU  =  Ilu(A  +  Bu) 


0  y4  nuBu  =  nu  A  Bu,  A  +  Su#u  =  2U(A  +  #„) 

£)  (nu^u)(nr5,,)  =  nu,  ,  Au  Bv  =  uu  Au  Bu, 

2UAU  +  2VBV  =  Zu,,C4u  +  .BO  =  S«C4«  +  5«) 

o)  suiic^4u,  „  c  lit,  su  Au,  v 

From  these  fundamental  propositions,  the  whole  theory  of  relations  is 
developed.  Though  Schroder  carries  this  much  further  than  Peirce,  the 
general  outlines  are  those  of  Peirce's  calculus.  Perhaps  the  most  inter- 
esting of  the  new  items  of  Schroder's  treatment  are  the  use  of  "matrices" 
in  the  form  of  the  two-dimensional  array  of  individual  binary  relatives, 
and  the  application  of  the  calculus  of  relatives  to  Dedekind's  theory  of 
"chains  ",  as  contained  in  Was  sind  und  was  sollen  die  Zahlen. 

Notable  contributions  to  the  Boole-Schroder  algebra  were  made  by 
Anton  Poretsky  in  his  three  papers,  Sept  lois  fondamentales  de  la  tkeorie 
des  egalites  logiques  (1899),  Quelques  lois  ulterieures  de  la  theorie  des  egalites 
logiques  (1901),  and  Theorie  des  non-egalites  logiques  (1904).  (With  his 
earlier  works,  published  in  Russian,  1881-87,  we  are  not  familiar.)  Poret- 
sky's  Law  of  Forms,  Law  of  Consequences,  and  Law  of  Causes  will  be 
given  in  Chapter  II.  As  Couturat  notes,  Schroder  had  been  influenced 
overmuch  by  the  analogies  of  the  algebra  of  logic  to  other  algebras,  and 
these  papers  by  Poretsky  outline  an  entirely  different  procedure  which, 
though  based  on  the  same  fundamental  principles,  is  somewhat  more 
"natural"  to  logic.  Poretsky  's  method  is  the  perfection  of  that  type  of 
procedure  adopted  by  Jevons  and  characteristic  of  the  use  of  the  Venn 
diagrams. 

The  work  of  Frege,  though  intrinsically  important,  has  its  historical 
interest  largely  through  its  influence  upon  Mr.  Bertrand  Russell.  Although 
the  Begriffsschrift  (1879)  and  the  Grundlagen  der  Arithmetik  (1884)  both 


The  Development  of  Symbolic  Logic  115 

precede  Schroder's  Vorlesungen,  Frege  is  hardly  more  than  mentioned 
there;  and  his  influence  upon  Peano  and  other  contributors  to  the  Formu- 
laire  is  surprisingly  small  when  one  considers  how  closely  their  ta*sk  is  re- 
lated to  his.  Frege  is  concerned  explicitly  with  the  logic  of  mathematics 
but,  in  thorough  German  fashion,  he  pursues  his  analyses  more  and  more 
deeply  until  we  have  not  only  a  development  of  arithmetic  of  unprecedented 
rigor  but  a  more  or  less  complete  treatise  of  the  logico-metaphysical  problems 
concerning  the  nature  of  number,  the  objectivity  of  concepts,  the  relations 
of  concepts,  symbols,  and  objects,  and  many  other  subtleties.  In  a  sense, 
his  fundamental  problem  is  the  Kantian  one  of  the  nature  of  the  judgments 
involved  in  mathematical  demonstration.  Judgments  are  analytic,  de- 
pending solely  upon  logical  principles  and  definitions,  or  they  are  synthetic. 
His  thesis,  that  mathematics  can  be  developed  wholly  by  analytic  judg- 
ments from  premises  which  are  purely  logical,  is  likewise  the  thesis  of 
Russell's  Principles  of  Mathematics.  And  Frege's  Grundgesefee  der  Arith- 
metik,  like  Principia  Mathematica,  undertakes  to  establish  this  thesis — for 
arithmetic — by  producing  the  required  development. 

Besides  the  precision  of  notation  and  analysis,  Frege's  work  is  important 
as  being  the  first  in  which  the  nature  of  rigorous  demonstration  is  suf- 
ficiently understood.  His  proofs  proceed  almost  exclusively  by  substitu- 
tion for  variables  of  values  of  those  variables,  and  the  substitution  of  defined 
equivalents.  Frege's  notation,  it  must  be  admitted  is  against  him:  it  is 
almost  diagrammatic,  occupying  unnecessary  space  and  carrying  the  eye 
here  and  there  in  a  way  which  militates  against  easy  understanding.  It  is 
probably  this  forbidding  character  of  his  medium,  combined  with  the 
unprecedented  demands  upon  the  reader's  logical  subtlety,  which  accounts 
for  th'e  neglect  which  his  writings  so  long  suffered.  But  for  this,  the  revival 
of  logistic  proper  might  have  taken  place  ten  years  earlier,  and  dated  from 
Frege's  Grundlagen  rather  than  Peano's  Formulaire. 

The  publication,  beginning  in  1894,  of  Peano's  Formulaire  de  Mathe- 
matiques  marks  a  new  epoch  in  the  history  of  symbolic  logic.  Heretofore, 
the  investigation  had  generally  been  carried  on  from  an  interest  in  exact 
logic  and  its  possibilities,  until,  as  Schroder  remarks,  we  had  an  elaborated 
instrument  and  nothing  for  it  to  do.  With  Peano  and  his  collaborators,  the 
situation  is  reversed:  symbolic  logic  is  investigated  only  as  the  instrument 
of  mathematical  proof.  As  Peano  puts  it:  m 

"  The  laws  of  logic  contained  in  what  follows  have  generally  been  found 

173  Formulaire,  i  (1901),  9. 


116  A  Survey  of  Symbolic  Logic 

by  formulating,  in  the  form  of  rules,  the  deductions  which  one  comes  upon 
in  mathematical  demonstrations." 

The  immediate  result  of  this  altered  point  of  view  is  a  new  logic,  no 
less  elaborate  than  the  old — destined,  in  fact,  to  become  much  more  elabo- 
rate— but  with  its  elaboration  determined  not  from  abstract  logical  con- 
siderations or  by  any  mathematical  prettiness,  but  solely  by  the  criterion 
of  application.  De  Morgan  had  said  that  algebraists 'and  geometers  live 
in  "a  higher  realm  of  syllogism":  it  seems  to  have  required  the  mathe- 
matical intent  to  complete  the  rescue  of  logic  from  its  traditional  inanities. 

The  outstanding  differences  of  the  logic  of  Peano  from  that  of  Peirce 
and  Schroder  are  somewhat  as  follows :  m 

(1)  Careful  enunciation  of  definitions  and  postulates,  and  of  possible 
alternative    postulates,    marking    an    increased    emphasis    upon    rigorous 
deductive  procedure  in  the  development  of  the  system. 

(2)  The  prominence  of  a  new  relation,  e,  the  relation  of  a  member  of  a 
class  to  the  class. 

(3)  The  prominence  of  the  idea  of  a  prepositional  function  and  of 
"formal  implication"  and   "formal  equivalence",   as  against   "material 
implication"  and  "material  equivalence". 

(4)  Recognition  of  the  importance  of  "existence"  and  of  the  properties 
of  classes,  members  of  classes,  and  so  on,  with  reference  to  their  "existence". 

(5)  The  properties  of  relations  in  general  are  not  studied,  and  "relative 
addition"  does  not  appear  at  all,  but  various  special  relations,  prominent 
in  mathematics,  are  treated  of. 

The  disappearance  of  the  idea  of  relation  in  general  is  a  real  loss,  not  a 
gain. 

(6)  The  increasing  use  of  substitution  (for  a  variable  of  some  value  in 
its  range)  as  the  operation  which  gives  proof. 

We  here  recognize  those  characteristics  of  symbolic  logic  which  have 
since  been  increasingly  emphasized. 

The  publication  of  Principia  Mathematica  would  seem  to  have  deter- 
mined the  direction  of  further  investigation  to  follow  that  general  direction 
indicated  by  the  work  of  Frege  and  the  Formulaire.  The  Principia  is  con- 
cerned with  the  same  topics  and  from  the  same  point  of  view.  But  we  see 
here  a  recognition  of  difficulties  not  suggested  in  the  Formulaire,  a  deeper 
and  more  lengthy  analysis  of  concepts  and  a  corresponding  complexity  of 
procedure.  There  is  also  more  attention  to  the  details  of  a  rigorous 
method  of  proof. 

174  All  these  belong  also  to  the  Logica  Mathematica  of  C.  Burali  Forti  (Milan,  1894). 


The  Development  of  Symbolic  Logic  117 

The  method  by  which  the  mathematical  logic  of  Principia  Mathematica 
is  developed  will  be  discussed,  so  far  as  we  can  discuss  it,  in  the  concluding 
section  of  Chapter  IV.  We  shall  be  especially  concerned  to  point  out  the 
connection,  sometimes  lost  sight  of,  between  it  and  the  older  logic  of  Peirce 
and  Schroder.  And  the  use  of  this  logic  as  an  instrument  of  mathematical 
analysis  will  be  a  topic  in  the  concluding  chapter. 


CHAPTER    II 
THE   CLASSIC,   OR   BOOLE-SCHRODER,   ALGEBRA   OF  LOGIC 

I.     GENERAL  CHARACTER  OF  THE  ALGEBRA.    THE  POSTULATES  AND 
THEIR  INTERPRETATION 

The  algebra  of  logic,  in  its  generally  accepted  form,  is  hardly  old  enough 
to  warrant  the  epithet  "classic".  It  was  founded  by  Boole  and  given  its 
present  form  by  Schroder,  who  incorporated  into  it  certain  emendations 
which  Jevons  had  proposed  and  certain  additions — particularly  the  relation 
"is  contained  in"  or  "implies" — which  Peirce  had  made  to  Boole's  system. 
It  is  due  to  Schroder's  sound  judgment  that  the  result  is  still  an  algebra, 
simpler  yet  more  powerful  than  Boole's  calculus.  Jevons,  in  simplifying 
Boole's  system,  destroyed  its  mathematical  form;  Peirce,  retaining  the 
mathematical  form,  complicated  instead  of  simplifying  the  original  calculus. 
Since  the  publication  of  Schroder's  Vorlesungen  iiber  die  Algebra  der  Logik 
certain  additions  and  improved  methods  have  been  offered,  the  most  notable 
of  which  are  contained  in  the  studies  of  Poretsky  and  in  Whitehead's  Uni- 
versal Algebra.1 

But  if  the  term  "classic"  is  inappropriate  at  present,  still  we  may 
venture  to  use  it  by  way  of  prophecy.  As  Whitehead  has  pointed  out, 
this  system  is  a  distinct  species  of  the  genus  "algebra",  differing  from  all 
other  algebras  so  far  discovered  by  its  non-numerical  character.  It  is 
certainly  the  simplest  mathematical  system  with  any  wide  range  of  useful 
applications,  and  there  are  indications  that  it  will  serve  as  the  parent  stem 
from  which  other  calculuses  of  an  important  type  will  grow.  Already  sev- 
eral such  have  appeared.  The  term  "classic"  will  also  serve  to  distinguish 
the  Boole-Schroder  Algebra  from  various  other  calculuses  of  logic.  Some 
of  these,  like  the  system*  of  Mrs.  Ladd-Franklin,  differ  through  the  use 
of  other  relations  than  +  ,  x ,  c ,'  and  = ,  and  are  otherwise  equivalent — 

1  For  Poretsky 's  studies,  see  Bibliography;  also  p.  114  above.  See  Whitehead's  Uni- 
versal Algebra,  Bk.  n.  Whitehead  introduced  a  theory  of  "discriminants"  and  a  treatment 
of  existential  propositions  by  means  of  umbral  letters.  This  last,  though  most  ingenious 
and  interesting,  seems  to  me  rather  too  complicated  for  use;  and  I  have  not  made  use  of 
"discriminants  ",  preferring  to  accomplish  similar  results  by  a  somewhat  extended  study  of 
the  coefficients  in  functions. 

118 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  119 

that  is  to  say,  with  a  "dictionary"  of  equivalent  expressions,  any  theorem 
of  these  systems  may  be  translated  into  a  theorem  of  the  Boole-Schroder 
Algebra,  and  vice  versa.  Others  are  mathematically  equivalent  as  far  as 
they  go,  but  partial.  And  some,  like  the  calculus  of  classes  in  Principia, 
Mathematica,  are  logically  but  not  mathematically  equivalent.  And, 
finally,  there  are  systems  such  as  that  of  Mr.  MacColl's  Symbolic  Logic 
which  are  neither  mathematically  nor  logically  equivalent. 

Postulates  for  the  classic  algebra  have  been  given  by  Huntington, 
by  Schroder  (in  the  Abriss],  by  Del  Re,  by  Sheffer  and  by  Bernstein.2  The 
set  here  adopted  represents  a  modification  of  Huntington's  third  set.d 
It  has  been  chosen  not  so  much  for  economy  of  assumption  as  for  "  natural- 
ness" and  obviousness. 

Postulated: 

A  class  K  of  elements  a,  b,  c,  etc.,  and  a  relation  x  such  that: 

1  •  1     If  a  and  b  are  elements  in  K,  then  a  x  b  is  an  element  in  K,  uniquely 
determined  by  a  and  6. 
1  •  2     For  any  element  a,  a  x  a  =  a. 
1  •  3     For  any  elements  a  and  b,  a  x  6  =  b  x  a. 
1  •  4     For  any  elements  a,  b,  and  c,  a  x  (b  x  c)  =  (a  x  6)  x  c. 
1  •  5     There  is  a  unique  element,  0,  in  K  such  that  a  x  0  =  0  for  every  ele- 
ment a. 
1 '  6    For  every  element  a,  there  is  an  element,  -a,  such  that  » 

1-61     If  x  x-a  =  0,  then  x  xa  =  x, 
and  1-62     If  y  x  a  =  y  and  y  x  -a  =  y,  then  y  =  0. 

The  element  1  and  the  relations  +  and  c  do  not  appear  in  the  above. 
These  may  be  defined  as  follows: 

1-7     1  =  -0        Def. 

1-8     a  +  b  =  -(-ax -6)         Def. 

1-9     a  c  b  is  equivalent  to  a  x6  =  a        Def. 

It  remains  to  be  proved  that  -a  is  uniquely  determined  by  a,  from 
which  it  will  follow  that  1  is  unique  and  that  a  +  b  is  uniquely  determined 
by  a  and  b. 

2  See  Bibl. 

3  See  "Sets  of  Independent  Postulates  for  the  Algebra  of  Logic",  Trans.  Amer.  Math. 
Soc.,  v  (1904),  288-309.     Our  set  is  got   by  replacing   +  in  Huntington's  set  by  x,  and 
replacing  the  second  half  of  G,  which  involves  1,  by  its  analogue  with  0.     Thus  1  can  be 
denned,  and  postulates  E  and  H  omitted.     Postulate  J  is  not  strictly  necessary. 


120  A  Survey  of  Symbolic  Logic 

The  sign  of  equality  in  the  above  has  its  usual  mathematical  meaning; 
i.  e.,  {  =  }  is  a  relation  such  that  if  x  =  y  and  $(x}  is  an  unambiguous 
function  in  the  system,  then  <p(x)  and  <p(y)  are  equivalent  expressions  and 
interchangeable.  It  follows  from  this  that  if  \f/(x)  is  an  ambiguous  function 
in  the  system,  and  x  =  y,  every  determined  value  of  $(x},  expressible  in 
terms  of  x,  is  similarly  expressible  in  terms  of  y.  Suppose,  for  example, 
that  -a,  "negative  of  a",  is  an  ambiguous  function  of  a.  Still  we  may  write 
-a  to  mean,  not  the  function  "negative  of  a"  itself,  but  to  mean  some 
(any)  determined  value  of  that  function — any  one  of  the  negatives  of  a — 
and  if  -a  =  b,  then  <p(-u)  and  <p(b)  will  be  equivalent  and  interchangeable. 
This  principle  is  important  in  the  early  theorems  which  involve  negatives. 

We  shall  develop  the  algebra  as  an  abstract  mathematical  system :  the 
terms,  a,  b,  c,  etc.,  may  be  any  entities  which  have  the  postulated  properties, 
and  x ,  + ,  and  c  may  be  any  relations  consistent  with  the  postulates. 
But  for  the  reader's  convenience,  we  give  two  possible  applications:  (1)  to 
the  system  of  all,  continuous  and  discontinuous,  regions  in  a  plane,  the 
null-region  included,  and  (2)  to  the  logic  of  classes.4 

(1) 

For  the  first  interpretation,  a  x  b  will  denote  the  region  common  to  a 
and  b  (their  overlapping  portion  or  portions),  and  a  +  b  will  denote  that 
which  is  either  a  or  b  or  both,  a  c  b  will  represent  the  proposition,  "Region 
a  is  completely  contained  in  region  b  (with  or  without  remainder)".  0  will 
represent  the  null-region,  contained  in  every  region,  and  1  the  plane  itself, 
or  the  "sum"  {  •»•  }  of  all  the  regions  in  the  plane.  For  any  region  a,  -a 
will  be  the  plane  except  a,  all  that  is  not-a.  The  postulates  will  then  hold 
as  follows: 

1-1  If  a  and  b  are  regions  in  the  plane,  the  region  common  to  a  and  b, 
ax  6,  is  in 'the  plane.  If  a  and  b  do  not  overlap,  then  ax6  is  the  null- 
region,  0. 

1  •  2     For  any  region  a,  the  region  common  to  a  and  a,  a  x  a,  is  a  itself. 
1  •  3     The  region  common  to  a  and  6  is  the  region  common  to  b  and  a. 
1  •  4     The  region  common  to  a  and  6  x  c  is  the  region  common  to  a  x  6 
and  c — is  the  region  common  to  all  three. 

1-5     The  region  common  to  any  region  a  and  the  null-region,  0,  is  0. 
1-6    For  every  region  a,  there  is  its  negative,  -a,  the  region  outside  or 
4  Both  of  these  interpretations  are  more  fully  discussed  in  the  next  chapter. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  121 

not  contained  in  a,  and  this  region  is  such  that 

1-61     If  -a  and  any  region  x  have  only  the  null-region  in 

common,  then  the  region  common  to  x  and  a  is  £  itself,  or  x  is  contained  in  a; 

and  1-62     If  the  region  common  to  y  and  a  is  y,  or  y  is  contained 

in  a,  and  the  region  common  to  y  and  -a  is  y,  or  y  is  contained  in  -a,  then  y 

must  be  the  null-region  which  is  contained  in  every  region. 

That  the  definitions,  1-7,  1-8,  and  1-9,  hold,  will  be  evident. 

(2) 

For  the  second  interpretation,  a,  b,  c,  etc.,  will  be  logical  classes,  taken 
in  extension — that  is,  a  =  b  will  mean  that  a  and  b  are  classes  composed  of 
identically  the  same  members,  a  x  &  will  represent  the  class  of  those 
things  which  are  members  of  a  and  of  b  both;  a  +  b,  those  things  which 
are  either  members  of  a  or  members  of  b  or  both,  a  c  b  will  be  the  proposi- 
tion that  all  members  of  a  are  also  members  of  b,  or  that  a  is  contained  in  b 
(with  or  without  remainder).  0  is  the  null-class  or  class  of  no  members; 
and  the  convention  is  required  that  this  class  is  contained  in  every  class. 
1  is  the  "universe  of  discourse"  or  the  class  which  contains  every  entity 
in  the  system.  For  any  class  a,  -a  represents  the  negative  of  a,  or  the  class 
of  all  things  which  are  not  members  of  a.  The  postulates  will  hold  as  fol- 
lows: 

1-1  If  a  and  b  are  logical  classes,  taken  in  extension,  the  members  com- 
mon to  a  and  b  constitute  a  logical  class.  In  case  a  and  b  have  no  members 
in  common,  this  class  is  the  null-class,  0. 

1-2     The  members  common  to  a  and  a  constitute  the  class  a  itself. 
1  •  3     The  members  common  to  a  and  b  are  the  same  as  those  common  to 
b  and  a. 

1  •  4  The  members  common  to  a,  b,  and  c,  all  three,  are  the  same,  whether 
we  first  find  the  members  common  to  b  and  c  and  then  those  common  to  a 
and  this  class,  or  whether  we  first  find  the  common  members  of  a  and  b 
and  then  those  common  to  this  class  and  c. 

1  •  5  The  members  common  to  any  class  a  and  the  null-class  are  none,  or 
the  null-class.  * 

1  •  6  For  every  class  a,  there  is  its  negative,  -a,  constituted  by  all  members 
of  the  "universe  of  discourse"  not  contained  in  a,  and  such  that: 

1-61     If  -a  and  any  class  x  have  no  members  in  common, 


122  A  Survey  of  Symbolic  Logic 

then  all  members  of  x  are  common  to  x  and  a,  or  x  is  contained  in  a; 

and  1-62     If  all  members  of  any  class  y  are  common  to  y  and  a, 
and  common  also  to  y  and  -a,  then  y  must  be  null. 

1-7     The  "universe  of  discourse",  "everything",  is  the  negative  of  the 
null-class,  "nothing". 

1-8     That  which  is  either  a  or  &  or  both  is  identical  with  the  negative  of 
that  which  is  both  not-a  and  not-6. 

1-9     That  "a  is  contained  in  6  "  is  equivalent  to  "The  class  a  is  identical 
with  the  common  members  of  a  and  b  ". 

That  the  postulates  are  consistent  is  proved  by  these  interpretations. 
In  the  form  given,  they  are  not  independent,  but  they  may  easily  be  made 
so  by  certain  alternations  in  the  form  of  statement.5 

The  following  abbreviations  and  conventions  will  be  used  in  the  state- 
ment and  proof  of  theorems : 

1.  axfe  will  generally  be  abbreviated  to  a  b  or  a -b,  ax(6xc)  to  a  (be}, 
ax-(fcx-c)  to  a-(6-c)  or  a--(6-c),  etc. 

2.  In  proofs,  we  shall  sometimes  mark  a  lemma  which  has  been  established 
as  (1),  or  (2),  etc.,  and  thereafter  in  that  proof  refer  to  the  lemma  by  this 
number.     Also,  we  shall  sometimes  write  "Q.E.D."  instead  of  repeating 
the  theorem  to  be  proved. 

3.  The  principles  (postulates,  definitions,  or  previous  theorems)  by  which 
any  step  in  proof  is  taken  will  usually  be  noted  by  a  reference  in  square 
brackets,  thus:    If  x  =  0,   then   [1-5]   ax  =  0.     Reference  to  principles 
whose  use  is  more  or  less  obvious  will  gradually  be  omitted  as  we  proceed. 
Theorems  will  be  numbered  decimally,   for  greater  convenience  in  the 
insertion  of  theorems  without  alteration  of  other  numbers.     The  non- 
decimal  part  of  the  number  will  indicate  some  major  division  of  theorems, 
as  1  •  indicates  a  postulate  or  definition.     Theorems  which  have  this  digit 
and  the  one  immediately  following  the  decimal  point  in  common  will  be 
different  forms  of  the  same  principle  or  otherwise  closely,  related. 

II.     ELEMENTARY  THEOREMS 
2-1     If  a  =  b,  then  a  c  =  b  c  and  c  a  =  c  b. 

This  follows  immediately  from  the  meaning  of  =  and  1-1. 
2-2     a  =  b  is  equivalent  to  the  pair,  acb  and  b  ca. 
If  a  =  b,  then  [1  -2]  a  b  =  a  and  b  a  =  b. 
6  On  this  point,  compare  with  Huntington's  set. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  123 

And  if  a  b  =  a  and  b  a  =  b,  then  [1  -3]  a  =  a  b  =  b  a  =  6. 
But  [1-9]  ab  =  a  is  equivalent  to  a  c  b  and  b  a  =  b  to  b  c  a. 
Equality  is,  then,  a  reciprocal  inclusion  relation. 
2-3     aca. 

a  =  a,  hence  [2-2]  Q.E.D. 
Every  element  is  "contained  in"  itself. 
2-4     a  -a  =  0  =  -a  a. 

[1-2]  a  a  —  a. 

Hence  [2-1,  1-4,  1  •  3]  a  -a  =  (a  a)  -a  =  a  (a  -a)  =  (a  -a)  a. 
Also  [1-2]  -a -a  =  -a.     Hence  a  -a  =  a  (-a  -a)  =  (a  -a)  -a. 
But  [1-62]  if  (a -a)  a  =  (a -a)  -a  =  a -a,  then  a -a  =  0. 
And  [1-3]  -a  a  =  a  -a.     Hence  also,  -a  a  =  0. 

Thus  the  product  of  any  element  into  its  negative  is  0,  and  0  is  the 
modulus  of  the  operation  x . 

2-5     a  -b  —  0  is  equivalent  to  a  b  =  a  and  to  a  c  b. 

If  a  b  =  a,    then    [1-4-5,    2-1-4]    a  -b  =  (a  6)  -6  =  a  (b  -b) 
=  a-0=0  (1) 

And  [1-61]  if  a  -6  =  0,  then  a  b  =  a  (2) 

By  (1)  and  (2),  a  -b  =  0  and  a  b  =  a  are  equivalent. 
And  [1  •  9]  a  b  =  a  is  a  c  b. 

We  shall  derive  other  equivalents  of  a  c  b  later.  The  above  is  required 
immediately.  In  this  proof,  we  have  written  "1-4-5"  and  "2-1-4" 
instead  of  "1-4,  1-5"  and  "2-1,  2-4".  This  kind  of  abbreviation  in 
references  will  be  continued. 

2-6     If  acO,  then  a  =  0. 

If  acO,  then  [1-9]  a-0  =  a.     But  [1-5]  a-0  =  0. 
2-7     If  a  c  b,  then  a  c  c  b  c>  and  c  a  c  c  b. 

If  acb,  then  [1-9]  a  6  =  a  and  [2-1]  (a6)c  =  ac  (1) 

But  [1-2-3-4]  (ab)c  =  (ba)c  =  b  (ac)  =  (ac)  b  =  [a  (cc)  &]  =  [(ac)  c]  6 

=  (ac)(c6)  =  (oc)(6c)  (2) 

Hence,  by  (1)  and  (2),  if  a  c  b,  then  (a  c)(6  c)  =  a  c  and  [1  -9]  a  c  c  6  c. 

And  [1-3]  c  a  =  a  c  and  cb  =  b  c.     Hence  also  c  a  c  c  6. 

2-8     -(-a)  =  a. 

[2-4]  -(-a) -a  =  0.     Hence  [2-5]  -(-a)  ca  (1) 

By  (1),  -[-(-a)]  c-o.     Hence  [2-7]  a --[-(-a)]  ca-a. 


124  A  Survey  of  Symbolic  Logic 

But  [2-4]  a  -a  =  0.     Hence  a --[-(-a)]  cO. 

Hence  [2-6]  a -[-(-a)]  =  0  and  [2-5]  a  c-(-a)  (2) 

[2-2]  (1)  and  (2)  are  equivalent  to  -(-a)  =  a. 
3-1     a  c  b  is  equivalent  to  -b  c  -a. 

[2-5]  a  c  6  is  equivalent  to  a  -6  =  0. 

And  [2-8]  a -6  =  -b  a  =  -b  -(-a). 

And  -6  -(-a)  =  0  is  equivalent  to  -6  c  -a. 

The  terms  of  any  relation  c  may  be  transposed  by  negating  both. 
If  region  a  is  contained  in  region  b,  then  the  portion  of  the  plane  not  in  b 
is  contained  in  the  portion  of  the  plane  not  in  a:  if  all  a's  are  6's,  all  non-6's 
are  non-a's.  This  theorem  gives  immediately,  by  2-8,  the  two  corollaries: 

3-12     a  c-6  is  equivalent  to  b  c-a;  and 
3-13     -a  c  6  is  equivalent  to  -b  c  a. 
3-2     a  =  6  is  equivalent  to  -a  =  -b. 

[2-2]  a  =  b  is  equivalent  to  (a  c6  and  b  ca). 
[3  •  1]  a  c  b  is  equivalent  to  -b  c  -a,  and  b  c  a  to  -a  c  -6. 
Hence  a  =  b  is  equivalent  to  (-a  c-6  and  -b  c-a),  which  is  equiva- 
lent to  -a  =  -b. 

The  negatives  of  equals  are  equals.     By  2-8,  we  have  also 
3-22     a  =  -b  is  equivalent  to  -a  =  b. 

Postulate  1-6  does  not  require  that  the  function  "negative  of"  be 
unambiguous.  There  might  be  more  than  one  element  in  the  system  having 
the  properties  postulated  of  -a.  Hence  in  the  preceding  theorems,  -a 
must  be  read  "any  negative  of  a  ",  -(-6)  must  be  regarded  as  any  one  of 
the  negatives  of  any  given  negative  of  b,  and  so  on.  Thus  what  has  been 
proved  of  -a,  etc.,  has  been  proved  to  hold  for  every  element  related  to  a 
in  the  manner  required  by  the  postulate.  But  we  can  now  demonstrate 
that  for  every  element  a  there  is  one  and  only  one  element  having  the 
properties  postulated  of  -a. 

3-3     -a  is  uniquely  determined  by  a. 

By  1-6,  there  is  at  least  one  element  -a  for  every  element  a. 
Suppose  there  is  more  than  one:  let  -ax  and  -«2  represent  any  two 
such. 

Then  [2-8]  -(-ai)  =  a  =  -(-a2).     Hence  [3-2]  -aj  =  -o2. 
Since  all  functions  in  the  algebra  are  expressible  in  terms  of  a,  b,  c,  etc., 
the  relation  x ,  the  negative,  and  0,  while  0  is  unique  and  a  x  b  is  uniquely 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  125 

* 

determined  by  a  and  b,  it  follows  from  3  •  3  that  all  functions  in  the  algebra 
are  unambiguously  determined  when  the  elements  involved  are  specified. 
(This  would  not  be  true  if  the  inverse  operations  of  "subtraction"  and 
"division"  were  admitted.) 

3-33     The  element  1  is  unique. 

[1-5]  0  is  unique,  hence  [3-3]  -0  is  unique,  and  [1-7]  1  =  -0. 
3-34    -1  =  0. 

[1-7]  1  =  -0.     Hence  [3-2]     Q.E.D. 

3  •  35  If  a  and  b  are  elements  in  K,  a  +  b  is  an  element  in  K  uniquely  deter- 
mined by  a  and  b. 

The  theorem  follows  from  3.3,  1-1,  and  1-8. 

3-37     If  a  =  b,  then  a  +  c  =  b  +  c  and  c  +  a  =  c  +  b. 

The  theorem  follows  from  3  •  35  and  the  meaning  of  = . 

3-4     -(a +  6)  =  -a-b. 

[1-8]  a  +  b  =  -(-a-b). 
Hence  [3-3,  2-8]  -(a  +  b)  =  -[-(-a-b)}  =  -a-b. 

3-41     -(a  b)  =  -a  +  -b. 

[1-8,  2-8]  -a  +  -b  =  -[-(-a) -(-b)]  =  -(ab). 

3-4  and  3-41  together  state  De  Morgan's  Theorem:  The  negative  of  a 
sum  is  the  product  of  the  negatives  of  the  summands;  and  the  negative  of  a 
product  is  the  sum  of  the  negatives  of  its  factors.  The  definition  1-8  is  a 
form  of  this  theorem.  Still  other  forms  follow  at  once  from  3«4  and  3-41, 
by  2- 8: 

3-42  -(-a  +  -b)  =  ab. 

3-43  -(a  +  -b)  =  -ab. 

3.44  -(-a  +  b)  =  a-b. 

3-45  -(a-b)  =  -a  +  b. 

3-46  -(-ab)  =  a  +  -b. 

From  De  Morgan's  Theorem,  together  with  the  principle,  3-2,  "The 
negatives  of  equals  are  equals",  the  definition  1-7,  1  =  -0,  and  theorem 
3-34,  -1  =  0,  it  follows  that  for  every  theorem  in  terms  of  x  there  is  a 
corresponding  theorem  in  terms  of  +  .  If  in  any  theorem,  each  element  be 
replaced  by  its  negative,  and  x  and  +  be  interchanged,  the  result  is  a 
valid  theorem.  The  negative  terms  can,  of  course,  be  replaced  by  positive, 


126  A  Survey  of  Symbolic  Logic 

since  we  can  suppose  x  =  -a,  y  =  -b,  etc.  Thus  for  every  valid  theorem 
in  the  system  there  is  another  got  by  interchanging  the  negatives  0  and  1 
and  the  symbols  x  and  +  .  This  principle  is  called  the  Law  of  Duality. 
This  law  is  to  be  illustrated  immediately  by  deriving  from  the  postulates 
their  correlates  in  terms  of  +  .  The  correlate  of  1  •  1  is  3  •  35,  already  proved. 

4-2     a  +  a  =  a. 

[l'2]-a-a  = -a.       Hence  [1-8,     3-2,    2-8]  a  +  a  =  -(-a-a)  = 
-(-a)  =  a. 

4-3     a  +  6  =  6  +  a. 

[1-3]  -a  -b  =  -b  -a.     Hence  [3-2]  -(-a  -6)  =  -(-b  -a). 
Hence  [1-8]  Q.E.D. 

4-4     a  +  (b  +  c)  =  (a  +  6)  +  c. 

[1-4]  -a(-fe-c)  =  (-a-fc)-c. 
Hence  [3-2]  -[-a  (-b  -c)]  =  -[(-a -6)  -c]. 
But  [3-46,  1-8]  -[-a(-fc-c)]  =  a  +  -(-b-c)  =  a+(6  +  c). 
And  [3-45,  1-8]  -[(-a-fc)-c]  =  -(-a-6)  +  c  =  (a  +  6)+c. 

4-5     a+1  =  1. 

[1-5]  -a-0  =  0.     Hence  [3-2]  -(-a-0)  =  -0. 
Hence  [3-46]  a  +  -0  =  -0,  and  [1-7]  a+1  =  1. 

4-61     If  -x  +  a  =  1,  then  x  a  =  x. 

If  -x  +  a  =  1,  then  [3-2-34-44]  x  -a  =  -(-z  +  a)  =-1=0. 
And  [2  •  5]  x  -a  =  0  is  equivalent  to  x  a  =  x. 

4  •  612     If  -x  +  a  =  1,  then  z  +  a  =  a. 

[4-61]  If  -a  +  a;  =  1,  then  ax  =  a,  and  [3-2]  -a  +  -x  =  -a      (1) 
By  (1)  and  2-8,  if  -x  +  a  =  1,  x  +  a  =  a. 

4-62     If  y  +  a  =  y  and  y  +  -a  =  y,  then  2/  =  1. 

If  T/  +  a  =  y,  [3-2]  -?/  -a  =  -(y  +  a)  =  -y. 
And  if  y  +  -a  =  y,  -y  a  =  -(y  +  -a)  =  ~y. 
But  [1-62]  if  -y  a  =  -y  and  -#  -a  =  -y,  -y  =  0  and  i/  =  -0  =  1. 

4-8     a  +  -a  =  1  =  -a  +  a.  (Correlate  of  2-4) 

[2-4]  -a  a  =  0.     Hence  [3-2]  a  +  -a  =  -(-a  a)  =-0  =  1. 
Thus  the  modulus  of  the  operation  +  is  1. 
4-9    -a  +  b  =  1,  a  +  b  =  b,  a  -b  =  0,  a  b  =  a,  and  a  c  6  are  all  equivalent. 

[2  •  5]  a  -b  =  0,  a  b  =  a,  and  a  c  6  are  equivalent. 
[3-2]  -a  +  b  =  1  is  equivalent  to  a -6  =  -(-a +  6)  =-1=0. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  127 

[4-612]  If  -a  +  b  =  1,  a  +  b  =  b. 

And  if  a  +  b  =  b,    [3  •  37]  -a  +  b  =  -a  +  (a  +  6)  =  (-a  +  a)  +  b  =  1+6 

=  1. 

Hence  a  +  b  =  b  is  equivalent  to  -a  +  b  =  1. 
We  turn  next  to  further  principles  which  concern  the  relation   c . 
5-1     If  a  c  b  and  6  cc,  then  ace. 

[1-9]  a  c  b  is  equivalent  to  a  b  =  a,  and  b  c  c  to  6  c  =  b. 
If  a  b  =  a  and  6  c  =  b,  a  c  =  (a  b)  c  =  a  (b  c)  =  a  b  =  a. 
But  ac  =  a  is  equivalent  to  ace. 

This  law  of  the  transitivity  of  the  relation  c  is  called  the  Principle  of 
the  Syllogism.  It  is  usually  included  in  any  set  of  postulates  for  the  algebra 
which  are  expressed  in  terms  of  the  relation  c  . 

5-2     a  b  ca  and  a  b  cb. 

(a  b)  a  =  a  (a  6)  =  (a  a)  b  =  a  b. 
But  (a  6)  a  —  a  b  is  equivalent  to  ab  ca. 
Similarly,  (a  b)  b  =  a  (b  b)  =  ab,  and  ab  cb. 

5-21     a  ca  +  b  and  b  ca  +  b. 

[5  •  2]  -a  -b  c  -a  and  -a  -b  c  -6.       » 
Hence  [3  •  12]  a  c  -(-a  -6)  and  be -(-a  -6). 
But  -(-a -6)  =  (a  +  b). 

Note  that  5-2  and  5-21  are  correlates  by  the  Law  of  Duality.     In 
general,  having  now  deduced  the  fundamental  properties  of  both  x  and  +  , 
we  shall  give  further  theorems  in  such  pairs. 
A  corollary  of  5-21  is: 

5-22     a  b  ca  +  b. 

[5-1-2- 21] 

5-3     If  a  c 6  and  c  c d,  then  ac  cb  d. 

[1-9]  If  acb  and  c  c  d,  then  a  b  =  a  and  c  d  =  c. 
Hence  (a  c)  (b  d)  =  (a  b)  (c  d)  =  a  c,  and  ac  cb  d. 

5-31     If  a  c  b  and  c  c  d,  then  a  +  c  c  b  +  d. 

If  acb  and  c  c  d,  [3-1]  -be  -a  and  -d  c  -c. 
Hence  [5-3]  -b-dc-a-c,  and  [3-1]  -(-a-c)  c-(-b-d). 
Hence  [1-8]  Q.E.D. 
By  the  laws,  a  a  =  a  and  a  +  a  =  a,  5-3  and  5-31  give  the  corollaries: 

5-32     If  a  c  c  and  b  c  c,  then  ab  cc. 


128  A  Survey  of  Symbolic  Logic 

5-33     If  a  c  c  and  b  c  c,  then  a  +  b  c  c. 

5-34    If  a  c  6  and  ace,  then  a  c  6  c. 

5-35     If  a  c  6  and  ace,  then  a  c  6  +  c. 

5-37    If  acb,  then  a  +  cc6  +  c.  (Correlate  of  2-7) 

[2-3]  ccc.     Hence  [5-31]  Q.E.D. 
5-4    a  +  ab  =  a. 

[5-21]  aca  +  ab  (1) 

[2-3]  a  ca,  and  [5-2]  a  b  ca.     Hence  [5-33]  a  +  a  6  ca  (2) 

[2-2]  If  (1)  and  (2),  then  Q.E.D. 

5-41     a  (a +  6)  =  a. 

[5-4]  -a  +  -a-6  =  -a.     Hence  [3-2]  -(-a  +  -a-&)  =  -(-a)  =  a. 
But  [3-4]  -(-a +  -«-&)  =  a--(-a-&)  =  a  (a +  6). 

5-4  and  5-41  are  the  two  forms  of  the  Law  of  Absorption.     We  have 
next  to  prove  the  Distributive  Law,  which  requires  several  lemmas. 

5*5    a  (6  +  c)  =  ab  +  ac. 

Lemma  I:    a  b  +  ac  ca  (b  +  c). 

[5-2]  ab  ca  and  ac  ca.     Hence  [5  •  33]  a  b  +  a  c  c  a  (1) 

[5-2]  ab  cb  and  ac  cc.     But  [5-21]  b  cb  +  c  and  c  cb  +  c. 
Hence  [5-1]  a  b  cb  +  c  and  a  c  cb  +  c. 

Hence  [5  •  33]  a  b  +  a  c  c  b  +  c  (2) 

[5-34]  If  (1)  and  (2),  then  a  b  +  a  c  ca  (b  +  c). 

Lemma  2 :     If  p  c  q  is  false,  then  there  is  an  element  x,  =}=  0,  such  that 
x  c  p  and  x  c  -q. 

p-q  is  such  an  element,  for  [5-2]  p-qcp  and  p-qc-q;    and 
[4-9]  if  p  -q  =  0,  then  p  c  q,  hence  if  p  c  q  is  false,  then  p  -q  =}=  0. 
(This  lemma  is  introduced  in  order  to  simplify  the  proof  of  Lemma  3.) 
Lemma  3 :     a  (b  +  c)  cb  +  ac. 

Suppose  this  false.     Then,  by  lemma  2,  there  is  an  element  x, 
4=  0,  such  that 

xca(b  +  c)  (1) 

and        x  c  -(b  +  ac) 

But  [?-12]  if  x  c-(b  +  a  c),  then  b  +  acc-x  (2) 

[5-1]  If  (1),  then  since  [5-2]  a  (b  +  ac)  ca,  x  ca  (3) 

and  also,  since  a(b  +  ac)  cb  +  c,  x  cb  +  c     »  (4) 

[5  •  1]  If  (2),  then  since  [5  •  21]  b  c  b  +  a  c,  b  c  -x  and  [3  •  12]  x  c  -b     (5) 
Also  [5  •  1]  if  (2),  then  since  [5  •  21]  a  c  c  b  +  a  c,  a  c  c  -x  and  [3  •  12] 
x  c  -(a  c)  (6) 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  129 

From  (6)  and  (3),  it  follows  that  x  cc  must  be  false;    for  if  x  cc 
and  (3)  x  c  a,  then  [5  •  34]  x  c  a  c.     But  if  x  c  a  c  and  (6)  x  c  -(a  c), 
then  [1-62]  x  =  0,  which  contradicts  the  hypothesis  x  ={=  0. 
But  if  x  c  c  be  false,  then  by  lemma  2,  there  is  an  element  y,  +  0, 
such  that 

ycx  (7) 

and        2/c-c,         or  [3-121         cc-?/  (8) 

.  [5  •  1]  If  (7)  and  (5),  then  y  c  -b  and  [3  •  12]  b  c  -y  (9) 

If  (8)  and  (9),  then  [5,-  33]  b  +  c  c  -y  and  [3  -12]yc  -(b  +  c)          (10) 
If  (7)  and  (4),  then  [5>l]ycb  +  c  (11) 

[1-9]  If  (11),  then  y  (b  +  c)  =  y,  and  if  (10),  y--(b  +  c)  =  y       (12) 
But  if   (12),  then  [1-62]  y  =  0,  which  contradicts  the  condition, 

*>o. 

Hence  the  supposition — that  a(b  +  c)  c  b  +  a  c  be  false — is  a  false 

supposition,  and  the  lemma  is  established. 
Lemma  4:     a  (b  +  c)  ca  b  +  a  c. 

By  lemma  3,  a  (6  +  c)  c  b  +  a  c. 
Hence  [2-7]  a  [a  (b  +  c)]  c  a  (b  +  a  c) . 
But  a  [a  (6  +  c)]  =  (a  a)(6  +  c)  =  a  (b  +  c). 
And  a  (b  +  a  c)  =  a  (a  c  +  6).     Hence  a  (6  +  c)  c  a  (a  c  +  6). 
But  by  lemma  3,  a  (a  c  +  6)  c  a  c  +  a  b. 
And  ac  +  ab  =  ab  +  ac.     Hence  a  (b  +  c)  c  a  b  +  a  c. 
Proof  of  the  theorem :     [2-2]  Lemma  1  and  lemma  4  are  together  equiva- 
lent to  a  (b  +  c)  =  a  b  +  a  c. 

This  method  of  proving  the  Distributive  Law  is  taken  from  Huntington, 
"Sets  of  Independent  Postulates  for  the  Algebra  of  Logic  ".  The  proof  of 
the  long  and  difficult  lemma  3  is  due  to  Peirce,  who  worked  it  out  for  his 
paper  of  1880  but  mislaid  the  sheets,  and  it  was  printed  for  the  first  time  in 
Huntington's  paper.6 

5-51     (a  +  b)(c  +  d)  =  (a  c  +  b  c)  +  (a  d  +  b  d). 

•    [5  •  5]  (a  +  6)  (c  +  d)  =  (a  +  6)  c  +  (a  +  6)  d  =  (a  c  +  b  c)  +  (a  d  +  b  d) . 

5-52     a  +  bc  =  (a  +  6)(a  +  c).  (Correlate  of  5 •  5) 

[5-51]  (a  +  6)  (a  +  c)  =  (a  a  +  b  a)  +  (a  c  +  b  c) 

=  [(a  +  a  6)  +  a  c]  +  &*e. 

But  [5-4]  (a  +  a  6)  +  a  c  =  a  +  a  c  =  a.     Hence  Q.E.D. 
Further  theorems  which  are  often  useful  in  working  the  algebra  and 
which  follow  readily  from  the  preceding  are  as  follows: 

6  See  "Sets  of  Independent  Postulates,  etc.",  loc.  c,it.,  p.  300,  footnote. 
10 


130  A  Survey  of  Symbolic  Logic 

5-6    a-1  =  a  =  1-a. 

[1-5]  a-0  =  0.     Hence  a--l  =  0. 
But  [1-61]  if  a--l  =  0,  then  a-1  =  a. 

5-61     acl. 

[1-9]  Since  a-1  =  a,  acl. 
5-62     a  +  0  =  a  =  0  +  a. 

-a--0  =  -a-1  =  -a.     Hence  [3-2]  a  +  0  =  -(-a--0)  =  -(-a)  =  a. 
5-63     Oca. 

0-a  =  a-0  =  0.     Hence  [1-9]  Q.E.D. 
5-64     1  ca  is  equivalent  to  a  =  1. 

[2 •  2]  a  =  1  is  equivalent  to  the  pair,  acl  and  lea. 
But  [5-61]  acl  holds  always.     Hence  Q.E.D. 

5-65     a  c  0  is  equivalent  to  a  =  0. 

[2  •  2]  a  =  0  is  equivalent  to  the  pair,  a  c  0  and  Oca. 
But  [5-63]  0  c  a  holds  always.     Hence  Q.E.D. 

5-7     If  a  +  6  =  a:  and  a  =  0,  then  b  =  x. 
If  a  =  0,  a  +  6  =  0  +  6  =  6. 

5-71     If  a  b  =  x  and  a  =  1,  then  b  =  re. 

If  a  =  1,  a  6  =  1-6  =  6. 
5-72     a  +  b  =  0  is  equivalent  to  the  two  equations,  a  =  0  and  6  =  0. 

If  a  =  0  and  6  =  0,  then  a  +  6  =  0  +  0  =  0. 
And  if  a  +  6  =  0,  -a  -6  =  -(a  +  6)  =  -0  =  1. 
But  if  -a -6  =  1,  a  =  a-1  =  a(-a -6)  =  (a -a)  -6  =  0--6  =  0. 
And  [5-7]  if  a  +  6  =  0  and  a  =  0,  then  6  =  0. 

5-73     a  6  =  1  is  equivalent  to  the  two  equations,  a  =  1  and  6  =  1. 

If  a  =  1  and  6  =  1,  then  a  6  =  1-1  =  1. 
And  if  a  6  =  1,  -a +  -6  =  -(a  6)  =-1=0.     Hence  [5-72]  -a  =  0 

and  -6  =  0. 

But  [3-2]  if  -a  =  0,  a  =  1,  and  if  -6  =  0,  6  =  1. 

5-7  and  5-72  are  important  theorems  of  the  algebra.  5-7,  "Any  null 
term  of  a  sum  may  be  dropped",  would  hold  in  almost  any  system;  but 
5-72,  "If  a  sum  is  null,  each  of  its  summands  is  null",  is  a  special  law 
characteristic  of  this  algebra.  It  is  due  to  the  fact  that  the  system  con- 
tains no  inverses  with  respect  to  +  and  0.  a  and  -a  are  inverses  with 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  131 

respect  to    x  and  0  and  with  respect  to   +    and  1.     5-71  and  5-73,  the 
correlates  of  5-7  and  5-72,  are  less  useful. 

5-8     a  (b  +  -b)  =  a  b  +  a  -b  =  a. 

[5-5]  a  (b  +  -6)  =  a  b  +  a  -b. 
And  [4-8]  6  +  -b  =  1.     Hence  a  (b  +  -b)  =  a- 1  =  a. 

5-85     a  +  b  =  a  +  -ab. 

[5-8]  b  =  ab  +  -ab. 

Hence  a  +  b  =  a  +  (a  b  +  -a  b)  =  (a  +  a  6)  +  a  -6. 
But  [5-4]  a  +  ab  =  a.     Hence  Q.E.D. 

It  will  be  convenient  to  have  certain  principles,  already  proved  for  two 
terms  or  three,  in  the  more  general  form  which  they  can  be  given  by  the 
use  of  mathematical  induction.  Where  the  method  of  such  extension  is 
obvious,  proof  will  be  omitted  or  indicated  only.  Since  both  x  and  + 
are  associative,  we  can  dispense  with  parentheses  by  the  definitions: 

5-901     a  +  b  +  c=(a  +  b)+c        Def. 

5-902     abc  =  (ab)c        Def. 

5-91     a  =  a  (b  +  -b)(c +  -c)(d  +  -d~) . .  . 

[5-8] 
5-92     1  =  (a  +  -a)(6  +  -6)(c  +  -c)... 

[4-8] 
5-93     a  =  a  +  ab  +  ac  +  ad+... 

[5-4] 

5-931     a  =  a  (a +  &)(«  + c)(a  + d).  .  . 
[5-41] 

5-94     a(b  +  c  +  d+...)=ab  +  ac  +  ad+... 

[5-5] 
5-941     a  +  bcd,..   =  (a  +  6)(a  +  c)(a  +  d). . . 

[5-52] 

5-95     -(a  +  b  +  c  +  .  .  . )  =  -a  -b  -c  .  .  . 

If  the  theorem  hold  for  n  terms,  so  that 

-(oi  +  a2  +  . .  .  +  an)  =  -«i  -a2  . . .  -an 
then  it  will  hold  f or  n  -\-  1  terms,  for  by  3  •  4, 

-[(ai  +  a2  +  . . .  +  on)  +  aB+1]  =  -(ai  +  a2  +  . . .  +  an)  --an-i-i 
And  [3-4]  the  theorem  holds  for  two  terms.     Hence  it  holds  for  any 
number  of  terms. 


132  A  Survey  of  Symbolic  Logic 

5  •  951     -(a  bed...)  =  -a  +  -b  +  -c  +  -d  +  . .  . 
Similar  proof,  using  3-41. 

5-96     1  =  a  +  b  +  c  +  . .  .  +  -a  -b  -c  .  . . 
[4-8,  5-951] 

5-97     a  +  6  +  c+...=Ois  equivalent  to  the  set,  a  =  0,  b  =  0,  c  =  0,  . .  . 
[5-72] 

5-971     abed...  =  1  is  equivalent  to  the  set,  a  =  1,  b  =  1,  c  =  1,  . . . 

[5-73] 
5-98     a -bed...  =  ab-ac-ad... 

[1-2]  a  a  a  a  . .  .  =  a. 

5-981     a+(b  +  c  +  d+...~)  =  (a  +  6)  +  (a  +  c)  +  (a  +  d)  +  . . . 
[4-2]  a  +  a  +  a+  . . .  =  a. 

The  extension  of  De  Morgan's  Theorem  by  5-95  and  5-951  is  especially 
important.  5-91,  5-92,  and  5-93  are  different  forms  of  the  principle  by 
which  any  function  may  be  expanded  into  a  sum  and  any  elements  not 
originally  involved  in  the  function  introduced  into  it.  Thus  any  expression 
whatever  may  be  regarded  as  a  function  of  any  given  elements,  even  though 
they  do  not  appear  in  the  expression,- — a  peculiarity  of  the  algebra.  5  •  92, 
the  expression  of  the  universe  of  discourse  in  any  desired  terms,  or  expansion 
of  1,  is  the  basis  of  many  important  procedures. 

The  theorems  5-91-5-981  are  valid  only  if  the  number  of  elements 
involved  be  finite,  since  proof  depends  upon  the  principle  of  mathematical 
induction. 

III.     GENERAL  PROPERTIES  OF  FUNCTIONS 

We  may  use  f(x),  $(x,  y),  etc.,  to  denote  any  expression  which  involves 
only  members  of  the  class  K  and  the  relations  x  and  +  .  The  further 
requirement  that  the  expression  represented  by  f(x)  should  involve  x  or 
its  negative,  -x,  that  $(x,  y)  should  involve  x  or  -x  and  y  or  -y,  is  unnecessary, 
for  if  x  and  -x  do  not  appear  in  a  given  expression,  there  is  an  equivalent 
expression  in  which  they  do  appear.  By  5-91, 

a  =  a  (x  +  -.r)  =  a  x  +  a  -x  =  (ax  +  a  -x) (y  +  -y) 

=  axy  +  ax-y  +  a-xy  +  a-x  -y,  etc. 

a  x  +  a  -x  may  be  called  the  expansion,  or  development,  of  a  with  reference 
to  x.  And  any  or  all  terms  of  a  function  may  be  expanded  with  reference 
to  x,  the  result  expanded  with  reference  to  y,  and  so  on  for  any  elements 
and  any  number  of  elements.  Hence  any  expression  involving  only  ele- 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  133 

ments  in  K  and  the  relations  x  and  +  may  be  treated  as  a  function  of 
any  elements  whatever. 

If  we  speak  of  any  a  such  that  x  =  a  as  the  "  value  of  x  ",  then  a  value 
of  x  being  given,  the  value  of  any  function  of  x  is  determined,  in  this  algebra 
as  in  any  other.  But  functions  of  x  in  this  system  are  of  two  types:  (1) 
those  whose  value  remains  constant,  however  the  value  of  x  may  vary,  and 
(2)  those  such  that  any  value  of  the  function  being  assigned,  the  value  of  x 
is  thereby  determined,  within  limits  or  completely.  Any  function  which 
is  symmetrical  with  respect  to  x  and  -x  will  belong  to  the  first  of  these 
classes;  in  general,  a  function  which  is  not  completely  symmetrical  with 
respect  to  x  and  -x  will  belong  to  the  second.  But  it  must  be  remembered, 
in  this  connection,  that  a  symmetrical  function  may  not  look  symmetrical 
unless  it  be  completely  expanded  with  reference  to  each  of  the  elements 
involved.  For  example, 

a  +  -a  b  +  -b 

is  symmetrical  with  respect  to  a  and  -a  and  with  respect  to  b  and  -b.  Ex- 
panding the  first  and  last  terms,  we  have 

a  (b  +  -6)  +  -a  b  +  (a  +  -a)  -b  =  a  b  +  a  -b  +  -a  b  +  -a  -b  =  1 

whatever  the  value  of  a  or  of  b.  Any  function  in  which  an  element,  x, 
does  not  appear,  but  into  which  it  is  introduced  by  expanding,  will  be 
symmetrical  with  respect  to  x  and  -x. 

The  decision  wrhat  elements  a  given  expression  shall  be  considered  a 
function  of  is,  in  this  algebra,  quite  arbitrary  except  so  far  as  it  is  deter- 
mined by  the  form  of  result  desired.  The  distinction  between  coefficients 
and  "variables"  or  "unknowns"  is  not  fundamental.  In  fact,  we  shall 
frequently  find  it  convenient  to  treat  a  given  expression  first  as  a  function- 
say — of  x  and  y,  then  as  a  function  of  z,  or  of  x  alone.  In  general,  coef- 
ficients will  be  designated  by  capital  letters. 

The  Normal  Form  of  a  Function. — Any  function  of  one  variable,  f(x), 
can  be  given  the  form 

A  x  +  B  -x 

where  A  and  B  are  independent  of  x.  This  is  the  normal  form  of  functions 
of  one  variable. 

6-1  Any  function  of  one  variable,  f(x},  is  such  that,  for  some  A  and  some  B 
which  are  independent  of  x, 

f(x}  =  A  x  +  B  -x 


134  A  Survey  of  Symbolic  Logic 

Any  expression  which  involves  only  elements  in  the  class  K  and 
the  relations  x  and  +  will  consist  either  of  a  single  term  —  a  single 
element,  or  elements  related  by  x  —  or  of  a  sum  of  such  terms.  Only 
four  kinds  of  such  terms  are  possible:  (1)  those  which  involve  x, 
(2)  those  which  involve  -x,  (3)  those  which  involve  both,  and  (4) 
those  which  involve  neither.7 

Since  the  Distributive  Law,  5-5,  allows  us  to  collect  the  coefficients 
of  x,  of  -x  and  of  (x  -a-),  the  most  general  form  of  such  an  expres- 
sion is 

p  x  +  q  -x  +  r  (x  -x)  +  s 

where  p,  q,  r,  and  s  are  independent  of  x  and  -x. 
But  [2-4]  r(x-x]  =  r-0  =  0. 
And  [5-9]  s  =  s  x  +  s  -x. 

Hence  p  x  +  q-x  +  r  (x  -x)  +  s  =  (p  +  s)  x  +  (q  +  s}  -x. 
Therefore,  A  =  p  +  s,  B  —  q  +  s,  gives  the  required  reduction. 
The  normal  form  of  a  function  of  n  -\-  1  variables, 


l,   XZ,    ...    Xn,  Xn+l 

may  be  defined  as  the  expansion  by  the  Distributive  Law  of 

f(Xi,  Xz,    ...    £„)•&»+!  +/  '(Xi,  a-2,    ...    Xn)  —  OTn+l 

where  /  and  /  '  are  each  some  function  of  the  n  variables,  x\,  x2,  ...  xn,  and 
in  the  normal  form.  This  is  a  "step  by  step"  definition;  the  normal  form 
of  a  function  of  two  variables  is  defined  in  terms  of  the  normal  form  of 
functions  of  one  variable;  the  normal  form  of  a  function  of  three  variables 
in  terms  of  the  normal  form  for  two,  and  so  on.8  Thus  the  normal  form 
of  a  function  of  two  variables,  $>(x,  y),  will  be  found  by  expanding 

(A  x  +  B  -x)  y+(Cx  +  D  -x)  -y 
It  will  be,  Axy  +  B-xy  +  Cx-y  +  D-x-y 

The  normal  form  of  a  function  of  three  variables,  ^?(x,  y,  z),  will  be 
A  x  y  z  +  B  -x  y  z  +  C  x-y  z  +  D  -x  -y  z  +  E  xy  -z  +  F  -x  y  -z 

+  G  x  -y  -z  +  H  -x  -y  -z 

And  so  on.     Any  function  in  the  normal  form  will  be  fully  developed  with 

7  By  a  term  which  "involves"  x  is  meant  a  term  which  either  is  x  or  has  x  "as  a 
factor".     But  "factor"  seems  inappropriate  in  an  algebra  in  which  h  x  is  always  contained 
in  x,  h  x  ex. 

8  This  definition  alters  somewhat  the  usual  order  of  terms  in  the  normal  form  of  func- 
tions.    But  it  enables  us  to  apply  mathematical  induction  and  thus  prove  theorems  of  a 
generality  not  otherwise  to  be  attained. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  135 

reference  to  each  of  the  variables  involved  —  that  is,  each  variable,  or  its 
negative,  will  appear  in  every  term. 

6-11     Any  function  may  be  given  the  normal  form. 

(a)  By  6-1,  any  function  of  one  variable  may  be  given  the 
normal  form. 

(6)  If  functions  of  n  variables  can  be  given  the  normal  form, 
then  functions  of  n  +  1  variables  can  be  given  the  normal  form,  for, 

Let  $(xi,  x2,  ...  xn,  xn+i)  be  any  function  of  n  +  1  variables. 
By  definition,  its  normal  form  will  be  equivalent  to 

f(Xi,  XZ,    ...    Xn)  'Xn+i  +  /  '(Xi,  Xz,    ...    Z»)  —Xn+l 

wLere  /  and  /  '  are  functions  of  Xi,  x2,  ...  xn  and  in  the  normal 

form. 

By  the  definition  of  a  function,  3>(xi,  xz,  ...  xn,  xn+i)  may  be  re- 

garded as  a  function  of  xn+\.     Hence,  by  6-1,  for  some  A  and  some 

B  which  are  independent  of  xn^  i 

$(&i,  Xz,    ...    Xn,  Xn+l)    =    A  Xn+l  +  B  -Xn+l 

Also,  by  the  definition  of  a  function,  for  some  /  and  some  /  ' 

A  =  f(xi,  Xz,  ...  xn) 
and     B  =  f  '(xl}  x2,  ...  xn) 
Hence,  for  some/  and/  '  which  are  independent  of  xn+i 

i,  Xz,    ...    Xn,  Xn+i)    = 


Therefore,  if  the  functions  of  n  variables,  /  and  /  ',  can  be  given  the 

normal  form,  then  $(x1}  xz,  .  .  .  xn,  xn+i)  can  be  given  the  normal  form. 

(c)   Since  functions  of  one  variable  can  be  given  the  normal  form, 

and  since  if  functions  of  n  variables  can  be  given  the  normal  form, 

functions  of  n  +  1  variables  can  be  given  the  normal  form,  therefore 

functions  of  any  number  of  variables  can  be  given  the  normal  form. 

The  second  step,  (6),  in  the  above  proof  may  seem  arbitrary.     That  it 

is  valid,  is  due  to  the  nature  of  functions  in  this  algebra. 

6-  12     For  a  function  of  n  variables,  $(xi,  x2,  ...  .rn),  the  normal  form  will 

be  a  sum  of  2"  terms,  representing  all  the  combinations  of  Xi,  positive  or 

negative,  with  x2,  positive  or  negative,   with  .  .  .  with  xn,  positive  or 

negative,  each  term  having  its  coefficient. 


136  A  Survey  of  Symbolic  Logic 

(a)  A  normal  form  function  of  one  variable  has  two  terms,  and 
by  definition  of  the  normal  form  of  functions  of  n  +  1  variables,  if 
functions  of  k  variables  have  2k  terms,  a  function  of  k  +  1  variables 
will  have  2*  +  2k,  or  2h+},  terms. 

(6)  A  normal  form  function  of  one  variable  has  the  further 
character  described  in  the  theorem;  and  if  normal  form  functions 
of  k  variables  have  this  character,  then  functions  of  A'  -f-  1  variables 
will  have  it,  since,  by  definition,  the  normal  form  of  a  function  of 
k  +  1  variables  will  consist  of  the  combinations  of  the  (k  +  l)st 
variable,  positive  or  negative,  with  each  of  the  combinations  repre- 
sented in  functions  of  k  variables. 

Since  any  coefficient  may  be  0,  the  normal  form  of  a  function  may  con- 
tain terms  which  are  null.  Where  no  coefficient  for  a  term  appears,  the 
coefficient  is,  of  course,  1.  The  order  of  terms  in  the  normal  form  of  a 
function  will  vary  as  the  order  of  the  variables  in  the  argument  of  the 
function  is  varied.  For  example,  the  normal  form  of  <b(x,  y)  is,  by  defini- 
tion, 

Axy  +  B-xy+Cx-y  +  D-x-y 

and  the  normal  form  of  ^(y,  x)  is 

Pyx  +  Q-yx  +  Ry-x  +  S-y-x 

Except  for  the  coefficients,  these  differ  only  in  the  order  of  the  terms  and 
order  of  the  elements  in  the  terms.     And  since  +  and   x  are  both  associa- 
tive and  commutative,  such  a  difference  is  not  material. 
6-15     Any  two  functions  of  the  same  variables  can  differ  materially  only 
in  the  coefficients  of  the  terms. 

The  theorem  follows  immediately  from  6-12. 

In  consequence  of  6-15,  we  can,  without  loss  of  generality,  assume 
that,  for  any  two  normal  form  functions  of  the  same  variables  w-ith  which 
we  may  be  concerned,  the  order  of  terms  and  the  order  of  variables  in  the 
arguments  of  the  functions  is  the  same.  And  also,  in  any  function  of 
n  +  T  variables,  $(#1,  x2,  ...  xn,  xn+i),  which  is  equated  to 


2,  ...  xn-xn+i  +         i,  x2,  ...  xn  --xn+i 

xn+i  may  be  any  chosen  one  of  the  n  +  1  variables.     The  convention  that 
it  is  always  the  last  is  consistent  with  complete  generality  of  the  proofs. 
6-17     The  product  of  any  two  terms  of  a  function  in  the  normal  form  is 
null. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  137 

By  6-12,  for  any  two  terms  of  a  function  in  the  normal  form, 
there  will  be  some  variable,  xn,  such  that  xn  is  positive  in  one  of 
them  and  negative  in  the  other;  since  otherwise  the  two  terms 
would  represent  the  same  combination  of  x\,  positive  or  negative, 
with  xz,  positive  or  negative,  etc.  Consequently,  the  product  of 
any  two  terms  will  involye  a  factor  of  the  form  xn  -xn,  and  will 
therefore  be  null. 

Unless  otherwise  specified,  it  will  be  presumed  hereafter  that  any  func- 
tion mentioned  is  in  the  normal  form. 

The  Coefficients  in  a  Function. — The  coefficients  in  any  function  can  be 
expressed  in  terms  of  the  function  itself. 

6-21     If/(x)  =  Ax  +  B-x,  then /(I)  =  A. 

For /(I)  =  A-1  +  B--1  =  A  +  B-Q  =  A. 

6-22     If  .f(x)  =  Ax  +  B-x,  then  /(O)  =  B. 

For/(0)  =  A-Q  +  B--Q  =  0  +  5-1  =  B. 

6-23    f(x)  =  /(!)•  a; +/(<))-*. 

The  theorem  follows  immediately  from  6-1,  6-21  and  6-22. 

These  laws,  first  stated  by  Boole,  are  very  useful  in  reducing  compli- 
cated expressions  to  normal  form.     For  example,  if 

V(x)  =  a  c  (d  x  +  -d  -x)  +  (c  +  x]  d 
reduction  by  any  other  method  would  be  tedious.     But  we  have 

^(1)  =  a  c  (d-l  +  -d-Q)  +  (c  +  1)  d  =  ac  d  +  c  d  +  d  =  d 
and     SF(0)  =  ac  (d-Q  +  -d- 1)  +  (c  +  0)  d  =  ac  -d  +  c  d 
Hence  the  normal  form  of  ^(x)  is  given  by 

^f(x)  =  dx  +  (ac-d  +  c  d)  -x 

Laws  analogous  to  6-23,  also  stated  by  Boole,  may  be  given  for  functions 
of  more  than  one  variable.     For  example, 

f(x,y)  =/(!,  l)-*y+/(0,  1)  -x  y+f(l,  0)-*-y+/(0,  0)  -x  -y 
and  $(x,  y,  z)  =  $>(!,  1,  1)  -x  y  z  +  $(0,  1,  1)  --x  y  z  +  <I>(1,  0,  1)  -x  -y  z 

+  $(0,  0, 1)  —x  -yz  +  $(1, 1,  0)  -x  y  -z  +  $(0,  1,  0)—xy-z 
+  $(1,  0,  0)  -x  -y  -z  +  $(0,  0,  0)  —x  -y  -z 

We  can  prove  that  this  method  of  determining  the  coefficients  extends  to 
functions  of  any  number  of  variables. 


138  A  Survey  of  Symbolic  Logic 

.Ti     .To     .T3  .  .  .      Xn  \ 

6-24     If  G  r  be  any  term  of  ¥(£i,  #2,  z3,  ...  xn),  then 

r  will  be  the  coefficient,  C. 

U,  U,  U,    .  .  .    U  J 

(a)  By  6  •  23,  the  theorem  holds  for  functions  of  one  variable. 

(6)  If  the  theorem  hold  for  functions  of  k  variables,  it  will  hold 
for  functions  of  k  +  1  variables,  for, 

By  6-11,  any  function  of  k  +  1  variables,  &(xi,  x2,  ...  xk,  xk+i), 
is  such  that,  for  some  /  and  some  / ', 


"t:::!}          .    <" 

i,  i,  ...  1,1     ,  fi,  i,  ...  il       ,Ji,  i,  ...  il 

And*o,o,...o,0}=/io)o,...o}-°+/io)o,...or1 


o,o,...o 


Therefore,  if  every  term  of  /  be  of  the  form 

1,  1,  ...  1  1      f    .TI    xz  ...    xk 


f 

'  1  0,  0,  ...  0  J      I  -T!  -xz  .  .  .  -xk 

then  every  term  of  $  in  which  a^+i  is  positive  will  be  of  the  form 
l,  1,  ...  ll      f    .T!    Xz...     xk 


and  the  coefficient  of  any  such  term  will  be  /  -j  r  ,  which, 

l_  u,  u,  ...  u  j 

*•>*->•••*-)    1 

oo       o1 

Vy      \J  j        .     ,     .        W  y 

And  similarly,  if  every  term  of  /  '  be  of  the  form 

fi,  i,  ...  11  r  x,  xz...  xk\ 

J     I  0,  0,  ...  0  j      l-xl  -xz  .  .  .  -xk  J 
then  every  term  of  $  in  which  .T/t+i  is  negative  will  be  of  the  form 

,  i,  ...  IT  r  x,  xz...  xk 


„ 

f  A 

0,  0,  ...  0  J      l-Xi  -.T2  .  .  .  -xk 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  139 


and  the  coefficient  of  any  such  term  will  be  / ' 

by(2)'is*{oo"'o'° 

\_  \J,    \J,     .   .   .     \J, 

Hence  every  term  of  <l>  will  be  of  the  form 

1,   1,    ...    1,   1  1        f     Zi     X2  .  .  .      Xk     Xk+i 

0,  0,  ...  0,  0  j      i  -Xi  -xz  .  .  .  -xk  -xk+i 

(c)  Since  the  theorem  holds  for  functions  of  one  variable,  and 
since  if  it  hold  for  functions  of  k  variables,  it  will  hold  for  functions 
of  k  +  1  variables,  therefore  it  holds  for  functions  of  any  number  of 
variables. 

For  functions  of  one  variable,  further  laws  of  the  same  type  as  6-23 — 
but  less  useful — have  been  given  by  Peirce  and  Schroder. 

If /(a?)  =  Ax  +  B-x: 
6-25    /(I)  =f(A+B)  =  f(-A+-B). 
6-26    /(O)  =  f(A-B)  =f(-A-B). 
6-27    f(A)  =A  +  B=  /(-£)  =  f(A-B)  =  f(A  +  -B) 


6-28    /(£)  =  A-B  =f(-A]  =f(-A-B)  =f(-A  +  B) 

=  /(l)-/(0)  =f(x]-f(-x). 
The  proofs  of  these  involve  no  difficulties  and  may  be  omitted. 

In  theorems  to  be  given  later,  it  will  be  convenient  to  denote  the  coef- 
ficients in  functions  of  the  form  $(xi,  x%,  .  .  .  xn}  by  A\,  AI,  AS,  .  . .  Azn, 
or  by  Ci,  C2,  Cs,  .  .  .,  etc.  This  notation  is  perfectly  definite,  since  the 
order  of  terms  in  the  normal  form  of  a  function  is  fixed.  If  the  argument 
of  any  function  be  (xi,  x2,  ...  xn),  then  any  one  of  the  variables,  Xk,  will 
be  positive  in  the  term  of  which  Cm  is  the  coefficient  in  case 

p-2k~l  <  m  ^  (p  +  l)^*-1 

where  p  =  any  even  integer  (including  0).  Otherwise  Xk  will  be  negative 
in  the  term.  Thus  it  may  be  determined,  for  each  of  the  variables  in  the 
function,  whether  it  is  positive  or  negative  in  the  term  of  which  Cm  is  the 
coefficient,  and  the  term  is  thus  completely  specified.  We  make  no  use 
of  this  law,  except  that  it  validates  the  proposed  notation. 

Occasionally  it  will  be  convenient  to  distinguish  the  coefficients  of  those 
terms  in  a  function  in  which  some  one  of  the  variables,  say  Xk,  is  positive 
from  the  coefficients  of  terms  in  which  Xk  is  negative.  We  shall  do  this 


140  A  Survey  of  Symbolic  Logic 

by  using  different  letters,  as  PI,  P2,  PS,  .  .  .  ,  for  coefficients  of  terms  in 
which  Xk  is  positive,  and  Qi,  Qz,  Qa,  ...  for  coefficients  of  terms  in  which  Xk 
is  negative.  This  notation  is  perfectly  definite,  since  the  number  of  terms, 
for  a  function  of  n  variables,  is  always  2",  the  number  of  those  in  which  Xk 
is  positive  is  always  equal  to  the  number  of  those  in  which  it  is  negative, 
and  the  distribution  of  the  terms  in  which  Xk  is  positive,  or  is  negative,  is 
determined  by  the  law  given  above. 

The  sum  of  the  coefficients,  AI  +  Az  +  A3  +  .  .  .,  will  frequently  be  indi- 
cated by  ^A  or  ^Ah',  the  product,  Ai-Az'A3-  .  .  .  by  H^4  or  11^. 

h  h 

Since  the  number  of  coefficients  involved  will  always  be  fixed  by  the  func- 
tion which  is  in  question,  it  will  be  unnecessary  to  indicate  numerically  the 
range  of  the  operators  ^  and  IJ  . 

The  Limits  of  a  Function.  —  The  lower  limit  of  any  function  is  the  prod- 
uct of  the  coefficients  in  the  function,  and  the  upper  limit  is  the  sum  of 
the  coefficients. 

6-3     A  BcAx  +  B-xcA  +  B. 


Hence  [1  -9]  A  B  c  A  x  +  B  -x. 

And  (A  x  +  B  -x)(A  +  B)  =  Ax  +  AB-x  +  ABx  +  B-x 

=  (AB  +  A)x  +  (AB  +  B)  -x. 
But  [5-4]  A  B  +  A  =  A,  and  A  B  +  B  =  B. 

Hence     (A  x  +  B  -x)(A  +  B)  =  Ax  +  B  -x,    and    [1-9]     A  x  +  B  -x 
cA  +  B. 

6-31    f(B)  C/(.T)  cf(A). 

[6-3  and  6-26,  6-27] 

6-32     If  the  coefficients  in  any  function,  F(XI,  x2,  ...  xn),  be  Cif  C2,  C3,  .  .  .  , 
then 


(a)  By  6  •  3,  the  theorem  holds  for  functions  of  one  variable. 

(6)  Let  &(xi,  x2,  ...  Xk,  Xk+i)  be  any  function  of  k  +  1  variables. 
By  6  •  1  1  ,  for  some  /  and  some  /  ', 
<£>(zi,  a?2,  ...  Xk,  afc+i)  =  f(xlt  x*,  .  .  .  x^-Xk+i 

+/'0ri,  xz,  ...  Xk)'-xk+i     (1) 

Since  this  last  expression  may  be  regarded  as  a  function  of  £jt+i  in 
which  the  coefficients  are  the  functions  /  and  /  ',  [6  •  3] 

z,  .  .  .  xk)  x/  '(xi,  xz,  ...  xk)  c  <i>(.ri,  xz,  .  .  .  xk, 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  141 

Let  AI{$],  Az{<&},  AS{$},  etc.,  be  here  the  coefficients  in  <£;  Ai{f}, 
AZ{/},  As{f},  etc.,  the  coefficients  in/;  and  A\{f'},  Az{f'},  As{f'}, 
etc.,  the  coefficients  in  / '. 
If  IU{/}  c/and  lUm  c/',  then  [6-3] 


and,  by  (1),  IU{/}  x  TLA  {/'}  c  *. 

But  since  (1)  holds,  any  coefficient  in  $  will  be  either  a  coefficient 

in  /  or  a  coefficient  in  /  ',  and  hence 


Hence  if  the  theorem  hold  for  functions  of  k  variables,  so  that 
zi,  32,  ...  %k)     and 


then        ^{$}  £$(&!,  ar2,  •  •  •  xk,  xk+i). 

Similarly,  since  (1)  holds,  [6-23]  $<=/+/'. 
Hence  if/c^>{/}  and  /'  c  ^A{ff},  then  [5-31] 


But  since  any  coefficient  in  $  is  either  a  coefficient  in  /  or  a  coef- 

ficientin/', 

Hence  $  c 

Thus  if  the  theorem  hold  for  functions  of  k  variables,  it  will 
hold  for  functions  of  k  +  1  variables. 

(c)  Since  the  theorem  holds  for  functions  of  one  variable,  and 
since  if  it  hold  for  functions  of  k  variables,  it  will  hold  for  functions 
of  k  +  1  variables,  therefore  it  holds  generally. 

As  we  shall  see,  these  theorems  concerning  the  limits  of  functions  are 
the  basis  of  the  method  by  which  eliminations  are  made. 

Functions  of  Functions.  —  Since  all  functions  of  the  same  variables  may 
be  given  the  same  normal  form,  the  operations  of  the  algebra  may  frequently 
be  performed  simply  by  operating  upon  the  coefficients. 

6-4     If  /Or)  =  A  x  +  B  -x,  then  -[f(x}\  =  -Ax  +  -B  -x. 
[3  -4]  -(A  x  +  B  -x)  =  -(A  x)  •  -(B  -x) 

=  (-A  +  -X)(-B  +  x)  =  -A-B  +  -AX  +  -B  -x 

-  (-A  -B  +  -4)  x  +  (-A  -B  +  -B}  -x 

But  [5-4]  -A  -B  +  -A  =  -A  and  -A  -B  +  -B  =  -B. 
Hence  -(A  x  +  B  -x)  =  -A  x  +  -B  -x. 


142  A  Survey  of  Symbolic  Logic 

6-41     The  negative  of  any  function,  in  the  normal  form,  is  found  by  re- 
placing each  of  the  coefficients  in  the  function  by  its  negative. 

(a)  By  6-4,  the  theorem  is  true  for  functions  of  one  variable. 
(6)  If  the  theorem  hold  for  functions  of  k  variables,  then  it  will 
hold  for  functions  of  k  +  1  variables. 

Let  F(XI,  Xz,  ...  xk,  xk+i)  be  any  function  of  k  +  1  variables. 
Then  by  6-11  and  3  •  2,  for  some  /  and  some  /  ', 


...  xk, 

+  /'(&!,  Xz,    ...    Xk)'-Xk+l] 

But  f(xi,  Xz,  ...  Xk)-Xk+i+f'(xi,  Xz,  ...  Xk)--Xk+i  may  be  regarded 
as  a  function  of 
Hence,  by  6-4, 

~[f(Xi,  X*,    •  •  •   X^'Xk+i  +/  '(Xi,  Xz,    ...   Xk}  '- 

-[f'(Xi,  Xz,    ... 


Hence  if  the  theorem  be  true  for  functions  of  k  variables,  so  that 
the  negative  of  /  is  found  by  replacing  each  of  the  coefficients  in  /  by 
its  negative  and  the  negative  of  /  '  is  found  by  replacing  each  of  the 
coefficients  in  /  '  by  its  negative,  then  the  negative  of  F  will  be 
found  by  replacing  each  of  the  coefficients  in  F  by  its  negative,  for, 
as  has  just  been  shown,  any  term  of 


z,  ...  xk, 
in  which  xk+i  is  positive  is  such  that  its  coefficient  is  a  coefficient  in 


and  any  term  of 

~[F(xi,  x2,  ...  x 

in  which  xk+i  is  negative  is  such  that  its  coefficient  is  a  coefficient  in 


(c)  Since  (a)  and  (6)  hold,  therefore  the  theorem  holds  generally. 
Since  a  difference  in  the  order  of  terms  is  not  material,  6-41  holds  not 
only  for  functions  in  the  normal  form  but  for  any  function  which  is  com- 
pletely expanded  so  that  every  element  involved  appears,  either  positive 
or  negative,  in  each  of  the  terms.  It  should  be  remembered  that  if  any 
term  of  an  expanded  function  is  missing,  its  coefficient  is  0,  and  in  the 
negative  of  the  function  that  term  will  appear  with  the  coefficient  1. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  143 

6-42     The  sum  of  any  two  functions  of  the  same  variables,  <b(xi,  Xz,  ...  £„) 
and  ^f(xi,  Xz,  ...£„),  is  another  function  of  these  same  variables, 

F(XI,  x2,  ...  xn), 

such  that  the  coefficient  of  any  term  in  F  is  the  sum  of  the  coefficients  of 
the  corresponding  terms  in  $  and  SK 

By   6-15,    $(xi,  Xz,  ...  Xn)   and   V(xi,  x2,  ...  xn)   cannot  differ 
except  in  the  coefficients  of  the  terms. 

Let  AI,  AZ,  As,  etc.,  be  the  coefficients  in  $;   BI,  BZ,  Bs,  etc.,  the 
coefficients  of  the  corresponding  terms  in  V.     For  any  two  such  cor- 

,.  .         Xi    Xz  .  .  .    xn  I  Xi    Xz  .  .  .    xn 

responding  terms,  Ak  1  r  and  Bk  H 

{_  -Xl  -Xz    .  .  .    -Xn  J  L  ~Xi  -Xz    .  .  .    ~Xn 

Xz    ...        Xn    I  Xl      Xz    .  .  .       XT( 

Bk 


—Xi  —Xz  .  •  •  —Xn  j  (_  —Xi  ~X;j  .  .  .   —Xn 

=  ( A       JM  I    Xl    Xz  •  •  •    Xn 

L  ™t£l  ~ic2    •   •   •    """*^n 

And  since  addition  is  associative  and  commutative,  the  sum  of  the 
two  functions  is  equivalent  to  the  sum  of  the  sums  of  such  corre- 
sponding terms,  pair  by  pair. 

6-43     The  product  of  two  functions  of  the  same  variables,  <f>(xi,  xz,  ...  xn) 
and  ty(xi,  Xz,  . . .  xn},  is  another  function  of  these  same  variables, 

F(Xi,  Xz,    .  .  .    Xn}, 

such  that  the  coefficient  of  any  term  in  F  is  the  product  of  the  coeffi- 
cients of  the  corresponding  terms  in  $  and  ty. 

{/>»  <"V»  -V  /v»  rvt  /y» 

Xi      Xz    .  .  .       Xn    I  J       Xi      Xz  .  .  .       Xn    I      , 

r  and  Bk~\  r  be  any  two 

—Xi  —Xz    .  .  .    —X'n  J  L  —Xi  —Xz  .  .  .    —Xn  J 

corresponding  terms  in  <J>  and  SK 

Xz  ...   Xn\    D   f   Xi   Xz  .  .  .   Xn 
X  D I 


—Xi  —Xz  •  •  •  ~Xn  J       L  —Xi  —Xz  •  .  •  —Xn 

r 

Xi  Xz  •  •  •  xn 

-Xi  -Xz  .  .  .  ~Xn 

By  6  •  15,  $  and  ty  do  not  differ  except  in  the  coefficients,  and  by 
6-17,  whatever  the  coefficients  in  the  normal  form  of  a  function,  the 
product  of  any  two  terms  is  null.  Hence  all  the  cross-products  of 
terms  in  $  and  S^  will  be  null,  and- the  product  of  the  functions  will 


144  A  Survey  of  Symbolic  Logic 

be  equivalent  to  the  sum  of  the  products  of  their  corresponding  terms, 

pair  by  pair. 

Since  in  this  algebra  two  functions  in  which  the  variables  are  not  the 
same  may  be  so  expanded  as  to  become  functions  of  the  same  variables, 
these  theorems  concerning  functions  of  functions  are  very  useful. 

IV.    FUNDAMENTAL  LAWS  OF  THE  THEORY  OF  EQUATIONS 

We  have  now  to  consider  the  methods  by  which  any  given  element 
may  be  eliminated  from  an  equation,  and  the  methods  by  which  the  value 
of  an  "unknown"  may  be  derived  from  a  given  equation  or  equations.  The 
most  convenient  form  of  equation  for  eliminations  and  solutions  is  the 
equation  with  one  member  0. 

Equivalent  Equations  of  Different  Forms.  —  If  an  equation  be  not  in  the 
form  in  which  one  member  is  0,  it  may  be  given  that  form  by  multiplying 
each  side  into  the  negative  of  the  other  and  adding  these  two  products. 

7-1     a  =  b  is  equivalent  to  a-b  +  -a  6  =  0. 

[2-2]  a  =  b  is  equivalent  to  the  pair,  a  c  b  and  b  c  a. 
[4-9]  a  cb  is  equivalent  to  a  -b  =  0,  and  b  c  a  to  -a  b  —  0. 
And  [5  •  72]  a  -b  =  0  and  -a  b  =  0  are  together  equivalent  to  a  -b 

+  -a  b  =  0. 
The  transformation  of  an  equation  with  one  member  1  is  obvious: 

7-12     a  =  1  is  equivalent  to  -a  =  0. 
[3-2] 

By  6-41,  any  equation  of  the  form  f(xi,  x2,  ...  #„)  =  1  is  reduced  to  the 
form  in  which  one  member  is  0  simply  by  replacing  each  of  the  coefficients 
in  /  by  its  negative. 

Of  especial  interest  is  the  transformation  of  equations  in  which  both 
members  are  functions  of  the  same  variables. 


7-13     If  &(xi,  xz,  ...  xn)  and  ^f(xi,  xz,  ...  xn)  be  any  two  functions  of  the 
same  variables,  then 


x-2,  .  .  .  xn 

is  equivalent  to  F(XI,  xz,  ...  xn)  =  0,  where  F  is  a  function  such  that 
if  A  i,  AZ,  A3,  etc.,  be  the  coefficients  in  <£,  and  BI,  BZ,  B3,  etc.,  be  the  coef- 
ficients of  the  corresponding  terms  in  ^,  then  the  coefficients  of  the  corre- 
sponding terms  in  F  will  be  (Ai  -Bi  +  -Ai  J5i),  (A2  -B2  +  -A2  BZ],  (AS-B3 
+  -A3B3),  etc. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  145 

By  7- 1,  $  =  ^  is  equivalent  to  (<i>  x-¥)  +  (-<f>  x  ¥)  =0. 
By  6-41,  -<J>  and  -^  are  functions  of  the  same  variables  as  <i>  and  ^. 
Hence,  by  6-43,  <J>  x-ty  and  -<£  x  ^  will  each  be  functions  of  these 
same  variables,  and  by  6-42,    ($  x-^)  +  (-$  x>J>)   will  also  be  a 
function  of  these  same  variables. 

Hence  $,  SF,  -<£,  -^,  $  x-SI>,  -<l>  x  SI>,  and  ($  x-^)  +  (-<£  x  ^)  are  all 
functions  of  the  same  variables  and,  by  6-15,  will  not  differ  except 
in  the  coefficients  of  the  terms. 

If  Ak  be  any  coefficient  in  <£,  and  Bk  the  corresponding  coefficient 
in  ^,  then  by  6-41,  the  corresponding  coefficient  in  -$  will  be  -Ak 
and  the  corresponding  coefficient  in  -ty  will  be  -Bk- 
Hence,   by  6-43,  the  corresponding  coefficient  in   $x->3>  will  be 
Ak  -Bk,  and  the  corresponding  coefficient  in  -$  x  ^  will  be  -AkBk. 
Hence,  by  6-42,  the  corresponding  coefficient  in  ($  x-SI>)  +  (-$  x  >J>) 
will  be  Ak  -Bk  +  -AkBk. 

Thus  (<£  x  -^)  +  (-<£  x  ^)  is  the  function  F,  as  described  above,  and 
the  theorem  holds. 

By  7-1,  for  every  equation  in  the  algebra  there  is  an  equivalent  equation 
in  the  form  in  which  one  member  is  0,  and  by  7  •  13  the  reduction  can  usually 
be  made  by  inspection. 

One  of  the  most  important  additions  to  the  general  methods  of  the 
algebra  which  has  become  current  since  the  publication  of  Schroder's  work 
is  Poretsky's  Law  of  Forms.9  By  this  law,  given  any  equation,  an  equiva- 
lent equation  of  which  one  member  may  be  chosen  at  will  can  be  derived. 

7-15     a  =  0  is  equivalent  to  t  =  a  -t  +  -a  t. 

If  a  =  0,  a-t  +  -at  =  Q—t+l-t  =  t. 
And  if  t  =  a-t  +  -at,  then  [7  •  1] 

(a  -t  +  -a  t)  -t  +  (a  t  +  -a  -t)  I  =  0  =  a-t  +  at  =  a 

Since  t  may  here  be  any  function  in  the  algebra,  this  proves  that  every 
equation  has  an  unlimited  number  of  equivalents.  The  more  general  form 
of  the  law  is : 

7-16     a  =  b  is  equivalent  to  t  =  (a  b  +  -a  -6)  t  +  (a  -b  +  -a  b)  -L 

[7  •  1]  a  =  b  is  equivalent  to  a  -b  +  -a  b  =  0. 
And  [6  •  4]  -(a  -b  +  -a  6)  =  ab  +  -a  -b. 
Hence  [7 -15]  Q.E.D. 
The  number  of  equations  equivalent  to  a  given  equation  and  expressible 

9  See  Sept  lois  fondamentales  de  la  theorie  des  egalites  logiques,  Chap.  I. 
11 


146  A  Survey  of  Symbolic  Logic 

in  terms  of  n  elements  will  be  half  the  number  of  distinct  functions  which 
can  be  formed  from  n  elements  and  their  negatives,  that  is,  2~n/2. 

The  sixteen  distinct  functions  expressible  in  terms  of  two  elements, 
a  and  b,  are: 

a,  -a,  b,  -b,  0  (i.  e.,  a -a,  b-b,  etc.),  1  (i.  e.,  a  +  -a,  b  +  -b,  etc.),  ab, 
a  -b,  -a  b,  -a  -b,  a  +  b,  a  +  -b,  -a  +  b,  -a  +  -b,  a  b  +  -a  -b,  and  a  -b  +  -a  b. 

In  terms  of  these,  the  eight  equivalent  forms  of  the  equation  a  =  b  are: 

a  =  b;  -a  =  -b;  0  =  a  -b  +  -a  b;  1  =  ab  +  -a  -b;  ab  =  a  +  b;  a -b 
=  -ab;  -a-b  =  -a  +  -b;  and  a  +  -b  =  -a  +  b. 

Each  of  the  sixteen  functions  here  appears  on  one  or  the  other  side  of  an 
equation,  and  none  appears  twice. 

For  any  equation,  there  is  such  a  set  of  equivalents  in  terms  of  the 
elements  which  appear  in  the  given  equation.  And  every  such  set  has 
what  may  be  called  its  "zero  member"  (in  the  above,  0  =  a-b  +  -ab) 
and  its  "whole  member"  (in  the  above,  1  =  ab  +  -a-b).  If  we  observe 
the  form  of  7-16,  we  shall  note  that  the  functions  in  the  "zero  member" 
and  "whole  member"  are  the  functions  in  terms  of  which  the  arbitrarily 
chosen  t  is  determined.  Any  t  =  the  t  which  contains  the  function  {  =  0} 
and  is  contained  in  the  function  {  =  1 } .  The  validity  of  the  law  depends 
simply  upon  the  fact  that,  for  any  t,  0  ct  c  1,  i.  e.,  t  =  l-t  +  Q--t.  It  is 
rather  surprising  that  a  principle  so  simple  can  yield  a  law  so  powerful. 

Solution  of  Equations  in  One  Unknown. — Every  equation  which  is  pos- 
sible according  to  the  laws  of  the  system  has  a  solution  for  each  of  the  un- 
knowns involved.  This  is  a  peculiarity  of  the  algebra.  We  turn  first  to 
equations  in  one  unknown.  Every  equation  in  x,  if  it  be  possible  in  the 
algebra,  has  a  solution  in  terms  of  the  relation  c  . 

7-2     A  x  +  B  -x  =  0  is  equivalent  to  B  c  x  c  -A. 

[5-72]  A  x  +  B  -x  =  0  is  equivalent  to  the  pair,  A  x  =  0  and 
B  -x  =  0. 

[4  •  9]  B  -x  =  0  is  equivalent  to  B  c  x. 
And  A  x  =  0  is  equivalent  to  x  -(-A)  =  0,  hence  to  x  c-A. 

7-21     A  solution  in  the  form  //  c  x  c  K  is  indeterminate  whenever  the  equa- 
tion which  gives  the  solution  is  symmetrical  with  respect  to  x  and  -x. 
First,  if  the  equation  be  of  the  form  A  x  +  A  -x  =  0. 

The  solution  then  is,  A  ex  c-A. 

But  if  A  x  +  A  -x  —  0,  then  A  =  A  (x  +  -x)  =  A  x  +  A  -x  =  0,  and 

-A  =  1. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  147 

Hence  the  solution  is  equivalent  to  Oczcl,  which  [5-61-63]  is 

satisfied  by  every  value  of  x. 

In  general,  any  equation  symmetrical  with  respect  to  x  and  -x 

which  gives  the  solution,  H  c  x  c  K,  will  give  also  H  c  -x  c  K. 

But  if  H  c  x  and  H  c  -x,  then  [4  •  9]  H  x  =  H  and  H  -x  =  H. 

Hence  [1- 62]  #  =  0. 

And  if  xcK  and  -xcK,  then  [5-33]  x  +  -xcK,  and  [4-8,  5-63] 

K  =  1. 

Hence  H  c  x  c  K  will  be  equivalent  to  0  c  x  c  1. 

It  follows  directly  from  7-21  that  if  neither  x  nor  -x  appear  in  an  equa- 
tion, then  although  they  may  be  introduced  by  expansion  of  the  functions 
involved,  the  equation  remains  indeterminate  with  respect  to  x. 

7  •  22     An  equation  of  the  form  A  x  +  B  -x  =  0  determines  x  uniquely  when- 
ever A  =  -B,  B  =  -A. 

[3-22]   A  =  -B  and  -A  =  B  are  equivalent;    hence  either  of 

these  conditions  is  equivalent  to  both. 

[7  •  21  A  x  +  B  -x  =  0  is  equivalent  to  B  c  x  c  -A. 

Hence  if  B  =  -A,  it  is  equivalent  to  B  ex  cB  and  to  -A  ex  c-A, 

and  hence  [2-2]  to  x  =  B  =  -A. 

In  general,  an  equation  of  the  form  A  x  +  B  -x  =  0  determines  x  be- 
tween the  limits  B  and  -A.  Obviously,  the  solution  is  unique  if,  and  only 
if,  these  limits  coincide;  a,nd  the  solution  is  wholly  indeterminate  only 
when  they  are  respectively  0  and  1,  the  limiting  values  of  variables  generally. 
7-221  The  condition  that  an  equation  of  the  form  A  x  +  B  -x  =  0  be  pos- 
sible in  the  algebra,  and  hence  that  its  solution  be  possible,  is  A  B  =  0. 

By  6-3,  ABcAx  +  B-x.     Hence  [5-65]  if  A  x  +  B  -x  =  0,  then 

AB  =  0. 

Hence  if  A  B  +  0,  then  A  x  +  B  -x  =  0  must  be  false  for  all  values 

of  x. 

And  A  x  +  B  -x  =  0  and  the  solution  B  ex  c  -A  are  equivalent. 
A  B  =  0  is  called  the  "equation  of  condition"  of  A  x  +  B  -x  =  0:  it  is 
a  necessary,  not  a  sufficient  condition.  To  call  it  the  condition  that  A  x 
+•  B  -x  =  0  have  a  solution  seems  inappropriate :  the  solution  B  ex  c  -A 
is  equivalent  to  A  x  +  B  -x  =  0,  whether  A  x  +  B  -x  =  0  be  true,  false,  or 
impossible.  The  sense  in  which  A  B  =  0  conditions  other  forms  of  the 
solution  of  A  x  +  B  -x  =  0  will  be  made  clear  in  what  follows. 

The  equation  of  condition  is  frequently  useful  in  simplifying  the  solution. 


148  A  Survey  of  Symbolic  Logic 

(In  this  connection,  it  should  be  borne  in  mind  that  A  B  =  0  follows  from 
A  x  +  B  -x  =  0.)  For  example,  if 

a  b  x  +  (a  +  6)  -x  =  0 

then  (a  +  6)  car  c-(a  &).     But  the  equation  of  condition  is 
a  b  (a  +  6)  =  a  b  =  0,        or,        -(a  6)  =  1 

Hence  the  second  half  of  the  solution  is  indeterminate,  and  the  complete 
solution  may  be  written 

a  +  b  ex 

However,  this  simplified  form  of  the  solution  is  equivalent  to  the  original 
equation  only  on  the  assumption  that  the  equation  of  condition  is  satisfied 
and  a  b  =  0. 

Again  suppose  ax  +  b  -x  +  c  =  0 

Expanding  c  with  reference  to  x,  and  collecting  coefficients,  we  have 

(a  +  c)  x  +  (b  +  c)  -x  =  0 
and  the  equation  of  condition  is 

(a  +  c)  (6  +  c)  =  ab  +  a  c  +  b  c  +  c  =  ab  +  c  =  0 
The  solution  is  b  +  c  c  x  c  -a  -c 

But,  by  5-72,  the  equation  of  condition  gives  c  =  0,  and  hence  -c  —  1. 

Hence  the  complete  solution  may  be  written 

« 

b  ex  c  -a 

But  here  again,  the  solution  b  ex  c-a  is  equivalent  to  the  original  equation 
only  on  the  assumption,  contained  in  the  equation  of  condition,  that  c  =  0. 

This  example  may  also  serve  to  illustrate  the  fact  that  in  any  equation 
one  member  of  which  is  0,  any  terms  which  do  not  involve  x  or  -x  may  be 
dropped  without  affecting  the  solution  for  x.  If  a  x  +  b  -x  +  c  =  0,  then 
by  5-72,  a  x  +  b  -x  =  0,  and  any  addition  to  the  solution  by  retaining  c  will 
be  indeterminat^.  All  terms  which  involve  neither  the  unknown  nor  its 
negative  belong  to  the  "symmetrical  constituent"  of  the  equation — to  be 
explained  shortly. 

Poretsky's  Law  of  Forms  gives  immediately  a  determination  of  x  which 
is  equivalent  to  the  given  equation,  whether  that  equation  involve  x  or  not. 

7-23     A  x  +  B  -x  =  0  is  equivalent  to  x  =  -A  x  +  B  -x. 
[7  •  15]  A  x  +  B  -x  =  0  is  equivalent  to 
x  =  (A  x  +  B  -x~)  -x  +  (-A  x  +  -B  -x)  x  =  B  -x  +  -A  x 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  149 

This  form  of  solution  is  also  the  one  given  by  the  method  of  Jevons.10 
Although  it  is  mathematically  objectionable  that  the  expression  which 
gives  the  value  of  x  should  involve  x  and  -x,  this  is  in  reality  a  useful  and 
logically  simple  form  of  the  solution.  It  follows  from  7-2  and  7-23  that 
x  —  -A  x  +  B  -x  is  equivalent  to  B  c  x  c  -A . 

Many  writers  on  the  subject  have  preferred  the  form  of  solution  in 
which  the  value  of  the  unknown  is  given  in  terms  of  the  coefficients  and  an 
undetermined  (arbitrary)  parameter.  This  is  the  most  "mathematical" 
form. 

7-24  If  A  B  =  0,  as  the  equation  A  x  +  B  -x  =  0  requires,  then  A  x 
+  B  -x  =  0  is  satisfied  by  x  =  B  -u  +  -A  u,  or  x  =  B  +  u  -A,  where  u  is 
arbitrary.  And  this  solution  is  complete  because,  for  any  x  such  that 
A  x  +  B  -x  =  0  there  is  some  value  of  u  such  that  x  =  B  -u  +  -A  u  =  B 
+  u-A. 

(a)  By  6  •  4,  if  x  =  B  -u  +  -A  u,  then  -x  =  -B  -u  +  A  u. 
Hence  if  x  =  B  -u  +  -A  u,  then 
A  x  +  B  -x  =  A  (B  -u  +  -B  u}  +  B  (-B  -u  +  Au) 

=  AB-u  +  ABu  =  AB 

Hence  if  A  B  =  0  and  x  =  B  -u  +  -A  u,  then  whatever  the  value 
of  u,  A  x  +  B  -x  =  0. 

(6)  Suppose  x  known  and  such  that  A  x  +  B  -x  =  0. 
Then  if  x  =  B  -u  +  -A  u,  we  have,  by  7  •  1, 

(B  -u  +  -A  u)  -x  +  (-B  -u  +  A  u)  x 

=  (A  x  +  -A  -x}  u+  (B  -x  +  -B  x)  -u  =  0 
The  condition  that  this  equation  hold  for  some  value  of  u  is,  by  7  •  221, 

(AX  +  -A  -x}(B  -X+-BX)  =  A-BX+-AB-X  =  o 

This  condition  is  satisfied  if  A  x  +  B  -x  =  0,  for  then 

A  (B  +  -B)  x  +  (A  +  -A}  B  -x  =  A  B  +  A  -£  x  +  -A  B  -x  =  0 

and  by  5 -72,  A  -B  x  +  -A  B  -x  =  0. 

(c)   If  A  B  =  0,  then  B  -u  +  -A  u  =  B  +  u  -A,  for: 
If  A  B  =  0,  then  A  B  u  =  0. 
Hence  B  -u  +  -A  u  =  B  -u  +  -A  (B  +  -B)  u  +  A  B  u 

=  B  -u  +  (A  +  -A)  B  u  +  -A-B  u  =  B  (-u  +  u)  +  -A  -B  u 

=  B  +  -A-BU. 

But  [5-85]  B  +  -A  -B  u  =  B  +  u  -A. 
10  See  above,  p.  77. 


150  A  Survey  of  Symbolic  Logic 

Only  the  simpler  form  of  this  solution,  x  =  B  +  u  -A,  will  be  used  hereafter. 
The  above  solution  can  also  be  verified  by  substituting  the  value  given 
for  x  in  the  original  equation.     We  then  have 

A  (B  -u  +  -A  u}  +  B  (-B  -u  +  A  u)  =  A  B  -u  +  A  B  u  =  A  B 

And  if  A  B  =  0,  the  solution  is  verified  for  every  value  of  u. 

That  the  solution,  x  =  B  -u  +  -A  u  =  B  +  u  -A,  means  the  same  as 
Bc.xc.-A,  will  be  clear  if  we  reflect  that  the  significance  of  the  arbitrary 
parameter,  u,  is  to  determine  the  limits  of  the  expression. 

If  u  =  0,  B  -u  +  -A  u  =  B  +  u  -A  =  B. 

If    u  =  1,    B-u  +  -Au  =  -A    and    B  +  u-A  =  B  +  -A.     But    when 

AB  =  0,  B  +  -A  =  -A  B  +  -A  =  -A. 

Hence  x  —  B  -u  +  -A  u  =  B  +  u  -A  simply  expresses  the  fact,  otherwise 
stated  by  B  ex  c-A,  that  the  limits  of  x  are  B  and  -A. 

The  equation  of  condition  and  the  solution  for  equations  of  the  form 
C  x  +  D  -x  =  1,  and  of  the  form  A  x  +  B  -x  =  C  x  +  D  -x,  follow  readily 
from  the  above. 

7  •  25     The  equation  of  condition  that  C  x  +  D  -x  =  1  is  C  +  D  =  1,  and  the 
solution  of  C  x  +  D  -x  =  1  is  -D  c  x  c  C. 

(a)  By  6-3,  Cx  +  D  -x  c  C  +  D. 

Hence  if  there  be  any  value  of  x  for  which  Cx  +  D-x  =  1,  then 
necessarily  C  +  D  =  1 . 

(6)  If  Cx  +  D-x  =  1,   then   [6-4]  -Cx  +  -D-x  =  0,  and  [7-2] 
-DcxcC. 

7-26     If   C  +  D  =  1,   then   the   equation    Cx  +  D-x  =  1    is    satisfied  by 
x  =  -D  +  uC,  where  u  is  arbitrary. 

Since   [6-4]   C  x  +  D  -x  =  1   is  equivalent  to  -C  x  +  -D  -x  =  0, 

and   C  +  D  =  1   is  equivalent  to  -C  -D  =  0,  the  theorem  follows 

from  7-24. 

7-27     If  A  x  +  B  -x  =  C  x  +  D  -x,  the  equation  of  condition  is 

(A-C  +  -A  C)(B-D  +  -BD)  =  0 
and  the  solution  is  B  -D  +  -B  D  c  x  c  A  C  +  -A  -C,  or 

x  =  B  -D  +  -B  D  +  u  (A  C  +  -A  -C),  where  u  is  arbitrary. 
By  7'  13,  A  x  +  B  -x  =  C  x  +  D  -x  is  equivalent  to 
(A-C  +  -ACx  +  B-D  +  -Bd-x  =  0. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  151 

Hence,  by  7-221,  the  equation  of  condition  is  as  given  above. 
And  by  7  •  2  and  7  •  24,  the  solution  is 

B-D  +  -BDcxc-(A-C  +  -A  C),        or 
x  =  B  -D  +  -B  D  +  u--(A  -C  +  -A  C),  where  u  is  arbitrary. 
And  [6-4]  -(A-C  +  -AC)  =  AC  +  -A-C. 

The  subject  of  simultaneous  equations  is  very  simple,  although  the 
clearest  notation  we  have  been  able  to  devise  is  somewhat  cumbersome. 

7-3  The  condition  that  n  equations  in  one  unknown,  A1x  +  Bl-x  =  0, 
A*x  +  B2  -x  =  0,  ...  Anx  +  Bn  -x  =  0,  may  be  regarded  as  simultaneous,  is 
the  condition  that 

£  (A*  Bk)  =  0 

h,  k 

And  the  solution  which  they  give,  on  that  condition,  is 

^Bkcxc  H-Ak 

k  k 

or  x  =  23  Bk  +  u-  II  -.4*,  where  u  is  arbitrary. 

k  k 

By    6-42    and    5-72,    Alx  +  B1  -x  =  0,     A2x  +  B2  -x  =  0,     ... 
Anx  +  Bn  -x  =  0,  are  together  equivalent  to 

(A1  +  A2  +  .  .  .  +  An)x  +  (B1  +  B2  +  .  .  .  +  Bn)  -x  =  0 
or  £  Ak  *  +  Z  Bk  -x  =  0 

k  k 

By  7  •  23,  the  equation  of  condition  here  is 

2  Akx^Bk.=  0 

k  k 

But  Z  Ak  x  Z  Bk  =  (A1  +  A2  +  .  .  .  +  An)(Bl  +  Bz  +  .  .  .  +  Bn) 

k  k 

=  A1  B1  +  A1  Bz  +  .  .  .  +  A1  Bn  +  A*  B1  +  A2  B2  +  .  .  .  +  A2  Bn 
+  A3Bl  +  A3B2+...+A3Bn+...+AnBl+...+AnBn 

=  ^(A»Bk}. 

h,  k 

And  by  7  •  2  and  7  •  24,  the  solution  here  is 


or 

k  k 

And  by  5-95,  -{  2  Ak]  =  JI  ~Ak. 

k  k 

It  may  be  noted  that  from  the  solution  in  this  equation,  nz  partial  solu 


152  A  Survey  of  Symbolic  Logic 

tions  of  the  form  Bh  ex  c-A'  can  be  derived,  for 

Bh  c  £  Bk        and        H  -A"  c -A*. 

k  k 

Similarly,  22n  —  1  partial  solutions  can  be  derived  by  taking  selections  of 
members  of  X^  Bk  an^  II  -Ak. 

k  k 

Symmetrical  and  Unsymmetrical  Constituents  of  Equations. — Some  of 
the  most  important  properties  of  equations  of  the  form  A  x  +  B  -x  =  0  are 
made  clear  by  dividing  the  equation  into  two  constituents — the  most 
comprehensive  constituent  which  is  symmetrical  with  respect  to  x  and  -x, 
and  a  completely  unsymmetrical  constituent.  For  brevity,  these  may 
be  called  simply  the  "symmetrical  constituent"  and  the  "unsymmetrical 
constituent ".  In  order  to  get  the  symmetrical  constituent  complete,  it 
is  necessary  to  expand  each  term  with  reference  to  every  element  in  the 
function,  coefficients  included.  Thus  in  A  x  +  B  -x  =  0  it  is  necessary  to 
expand  the  first  term  with  respect  to  B,  and  the  second  with  respect  to  A. 

A(B  +  -B)x  +  (A  +  -A)  B  -x  =  A  B  x  +  A  B  -x  +  A  -B  x  +  -A  B  -x  =  0 
By  5  •  72,  this  is  equivalent  to  the  two  equations, 

A  B  (x  +  -x)  =  AB  =  0        and        A  -B  x  +  -A  B  -x  =  0 

The  first  of  these  is  the  symmetrical  constituent;  the  second  is  the  unsym- 
metrical constituent.  The  symmetrical  constituent  will  always  be  the  equa- 
tion of  condition,  while  the  unsymmetrical  constituent  will  give  the  solution. 
But  the  form  of  the  solution  will  most  frequently  be  simplified  by  con- 
sidering the  symmetrical  constituent  also.  The  unsymmetrical  constituent 
will  always  be  such  that  its  equation  of  condition  is  satisfied  a  priori.  Thus 
the  equation  of  condition  of 

A  -B  x .+  -A  B-x  =  0 

is  (A  -B)(-A  B)  =  0,  which  is  an  identity. 

By  this  method  of  considering  symmetrical  and  unsymmetrical  con- 
stituents, equations  which  are  indeterminate  reveal  that  fact  by  having 
no  unsymmetrical  constituent  for  the  solution.  Also,  the  method  enables 
us  to  treat  even  complicated  equations  by  inspection.  Remembering  that 
any  term  in  which  neither  x  nor  -x  appears  belongs  to  the  symmetrical 
constituent,  as  does  also  the  product  of  the  coefficients  of  x  and  -x,  the 
separation  can  be  made  directly.  For  example, 

(c  +  x)  d  +  -c  -d  +  (-a  +  -a;)  6  =  0 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  153 

will  have  as  its  equation  of  condition 

c  d  +  -c  -d  +  -ab  +  b  d  =  0 
and  the  solution  will  be 

bc.xc.-d 

Also,  as  we  shall  see  shortly,  the  symmetrical  constituent  is  always  the 
complete  resultant  of  the  elimination  of  x. 

The  method  does  not  readily  apply  to  equations  which  do  not  have 
one  member  0.  But  these  can  always  be  reduced  to  that  form.  How  it 
extends  to  equations  in  more  than  one  unknown  will  be  clear  from  the 
treatment  of  such  equations. 

Eliminations. — The  problem  of  elimination  is  the  problem,  what  equa- 
tions not  involving  x  or  -x  can  be  derived  from  a  given  equation,  or  equa- 
tions, wrhich  do  involve  x  and  -x.  In  most  algebras,  one  term  can,  under 
favorable  circumstances,  be  eliminated  from  two  equations,  two  terms 
from  three,  n  terms  from  n  +  1  equations.  But  in  this  algebra  any  number 
of  terms  (and  their  negatives)  can  be  eliminated  from  a  single  equation; 
and  the  terms  to  be  eliminated  may  be  chosen  at  will.  The  principles 
whereby  such  eliminations  are  performed  have  already  been  provided  in 
theorems  concerning  the  equation  of  condition. 

7-4  A  B  =  0  contains  all  the  equations  not  involving  x  or  -x  which  can 
be  derived  from  A  x  +  B  -x  =  0. 

By  7  •  24,  the  complete  solution  of  A  x  +  B  -x  =  0  is 

x  =  B  -u  +  -A  u 

Substituting  this  value  of  x  in  the  equation,  wre  have 
A  (B  -u  +  -A  u)  +  B  (-B  -u  +  A  u)  =  A  B  -u  +  A  B  u  =  A  B  =  0 

Hence  A  B  =  0  is  the  complete  resultant  of  the  elimination  of  x. 
It  is  at  once  clear  that  the  resultant  of  the  elimination  of  x  coincides 
with  the  equation  of  condition  for  solution  and  with  the  symmetrical  con- 
stituent of  the  equation. 

7-41  If  n  elements,  Xi,  x2,  xs,  ...  xn,  be  eliminated  from  any  equation, 
F(XI,  Xz,  xs,  ...  Xn)  =  0,  the  complete  resultant  is  the  equation  to  0  of  the 
product  of  the  coefficients  in  F(XI,  x2,  x3,  ...  xn). 

(a)  By  6-1  and  7-4,  the  theorem  is  true  for  the  elimination  of 
one  element,  x,  from  any  equation,  f(x)  =  0. 

(6)  If  the  theorem  hold  for  the  elimination  of  k  elements,  x\,  xz, 


154  A  Survey  of  Symbolic  Logic 

...  Xk,  from  any  equation,  $(x\,  Xz,  ...  Xk)  =  0,  then  it  will  hold 
for  the  elimination  of  k  +  1  elements,  x\,  x2,  ...  Xk,  Xk+i,  from  any 
equation,  ty(xi,  xz,  ...  Xk,  Xk+i)  =  0,  for: 

By  6-  11,  V(xi,  xz,  .  .  .  xk,  ar/fc+i)  =  f(xi,  Xz,  ...  xk}  -xk+i 


And  the  coefficients  in  SF  will  be  the  coefficients  in  /  and  /  '.  By 
7-4,  the  complete  resultant  of  eliminating  xk+i  from 

f(xi,  xz,  ...  Xk)  -xk+i  +/  r(xi,  x2,  ...  Xk)--xk+i  =  0 
is  /Oi,  x2,  ...  Xk)  x/  '(xi,  xz,  .  .  .  xk)  =  0 

And  by  6-43,  f(xi,  x2,  ...  xk]  *f'(xi,  x2,  ...  xk)  =  0  is  equivalent 
to  Q(XI,  x2,  ...  Xk)  =  0,  where  $  is  a  function  such  that  if  the 
coefficients  in/  be  PI,  P2,  P3,  etc.,  and  the  corresponding  coefficients 
in/'  be  Qi,  Qz,  Qs,  etc.,  then  the  corresponding  coefficients  in  $  will 
be  PiQi,  PzQz,  PsQs,  etc.  Hence  if  the  theorem  hold  for  the  elimina- 
tion of  k  elements,  x\,  x2,  ...  Xk,  from  3>(xi,  x2,  ...  xk)  =  0,  this 
elimination  will  give 

(P1Q1)(P2Q2)(P3Q3).  .  .  ==  (PiP2P3.  ..QiQ&.  .  .)  =  0, 

where  PiP2P3.  .  .QiQzQs-  •  •  is  the  product  of  the  coefficients  in  <£, 
or  in/  and/  '  —  i.  e.,  the  product  of  the  coefficients  in  ^. 
Hence  if  the  theorem  hold  for  the  elimination  of  k  elements,  x\,  xz, 
...  Xk,  from  Q(XI,  Xz,  ...  xk)  =  0,  it  will  hold  for  the  elimination  of 
k  +  1  elements,  xi,  xz,  ...  xk,  xk+\,  from  ^(a;i,  Xz,  ...  Xk,  xk+i)  =  0, 
provided  Xk+\  be  the  first  eliminated. 

But  since  the  order  of  terms  in  a  function  is  immaterial,  and  for 
any  order  of  elements  in  the  argument  of  a  function,  there  is  a 
normal  form  of  the  function,  ^+i  in  the  above  may  be  any  of  the 
A:  +  1  elements  in  ^,  and  the  order  of  elimination  is  immaterial. 

(c)  Since  (a)  and  (6)  hold,  therefore  the  theorem  holds  for  the 
elimination  of  any  number  of  elements  from  the  equation  to  0  of 
any  function  of  these  elements. 

By  this  theorem,  it  is  possible  to  eliminate  simultaneously  any  number  of 
elements  from  any  equation,  by  the  following  procedure:  (1)  Reduce  the 
equation  to  the  form  in  which  one  member  is  0,  unless  it  already  have  that 
form;  (2)  Develop  the  other  member  of  the  equation  as  a  normal-form 
function  of  the  elements  to  be  eliminated;  (3)  Equate  to  0  the  product  of 
the  coefficients  in  this  function.  This  will  be  the  complete  elimination 
resultant. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  155 

Occasionally  it  is  convenient  to  have  the  elimination  resultant  in  the 
form  of  an  equation  with  one  member  1,  especially  if  the  equation  which 
gives  the  resultant  have  that  form. 

7-42  The  complete  resultant  of  eliminating  n  elements,  x\,  x%,  ...  xn, 
from  any  equation,  F(x\,  x2,  ...  xn)  =  1,  is  the  equation  to  1  of  the  sum 
of  the  coefficients  in  F(XI,  x2,  ...  #„). 

Let  AI,  A2,  AS,  etc.,  be  the  coefficients  in  F(XI,  x2,  ...  xn). 
F(XI,  x2,  ...  Xn)  =  1  is  equivalent  to  -[F(xi,  x2,  ...  xn}\  =  0.  And 
by  6-41,  -[F(xi,  x2,  ...  #„)]  is  a  function,  $(xi,  x2,  ...  xn),  such 
that  if  any  coefficient  in  F  be  Ak,  the  corresponding  coefficient  in  $ 
will  be  -Ak. 

Hence,  by  7-41,  the  complete  resultant  of  eliminating  xi}  xz,  ...  xn, 
from  F(XI,  Xz,  ...  Xn)  =  1  is 

IL-A=0,        or        -{11-4}  =1 
But  [5-95]  -{  11-^1  =  T.A.     Hence  Q.E.D. 

For  purposes  of  application  of  the  algebra  to  ordinary  reasoning,  elimina- 
tion is  a  process  more  important  than  solution,  since  most  processes  of 
reasoning  take  place  through  the  elimination  of  "middle"  terms.  For 
example : 

If  all  6  is  x,     6  ex,     b  -x  =  0 

and  no  a  is  x,  a  x  =  0, 

then  a  x  +  b  -x  =  0.  Whence,  by  elimination,  a  b  =  0,  or  no  a  is  b. 
Solution  of  Equations  in  more  than  one  Unknown. — The  complete  solu- 
tion of  any  equation  in  more  than  one  unknown  may  be  accomplished  by 
eliminating  all  the  unknowns  except  one  and  solving  for  that  one,  repeating 
the  process  for  each  of  the  unknowns.  Such  solution  will  be  complete 
because  the  elimination,  in  each  case,  will  give  the  complete  resultant  which 
is  independent  of  the  unknowns  eliminated,  and  each  solution  will  be  a 
solution  for  one  unknown,  and  complete,  by  previous  theorems.  How- 
ever, general  formulae  of  the  solution  of  any  equation  in  n  unknowns,  for 
each  of  the  unknowns,  can  be  proved. 

7  •  5  The  equation  of  condition  of  any  equation  in  n  unknowns  is  identical 
with  the  resultant  of  the  elimination  of  all  the  unknowns;  and  this  resultant 
is  the  condition  of  the  solution  with  respect  to  each  of  the  unknowns  sepa- 
rately. 

(a)  If  the  equation  in  n  unknowns  be  of  the  form 

F(XI,  xz,  .  .  .  Xn)  =  0: 


156  A  Survey  of  Symbolic  Logic 

Let  the  coefficients  in  F(x\,  xz,  ...  #„)  be  A\,  At,  As,  etc.  Then, 
by  6 -32, 

II A  c  F(xi,  xz,  . . .  xn} 

and  [5  •  65]  H  A  =  0  is  a  condition  of  the  possibility  of 
F(XI,  xz,  ...  xn)  =  0 

And  [7-41]  II  A  =  0  is  the  resultant  of  the  elimination  of  x\,  Xz, 
...  xn,  from  F(XI,  x2)  ...  xn)  =  0. 

(6)  If  the  equation  in  n  unknowns  have  some  other  form  than 
F(XI,  x2,  ...  Xn)  =  0,  then  by  7-1,  it  has  an  equivalent  which  is 
of  that  form,  and  its  equation  of  condition  and  its  elimination 
resultant  are  the  equivalents  of  the  equation  of  condition  and 
elimination  resultant  of  its  equivalent  which  has  the  form 

F(Xi,  Xz,    ...    Xn)    =    0 

(c)  The  result  of  the  elimination  of  all  the  unknowns  is  the 
equation  of  condition  with  respect  to  any  one  of  them,  say  Xk, 
because : 

(1)  The  equation  to  be  solved  for  Xk  will  be  the  result  of  eliminat- 
ing all  the  unknowns  but  Xk  from  the  original  equation;  and 

(2)  The  condition  that  this  equation,  in  which  Xk  is  the  only 
unknown,  have  a  solution  for  Xk  is,  by  (a)  and  (6),  the  same  as  the 
result  of  eliminating  Xk  from  it. 

Hence  the  equation  of  condition  with  respect  to  Xk  is  the  same  as 
the  result  of  eliminating,  from  the  original  equation,  first  all  the 
other  unknowns  and  then  Xk- 

And  by  7-41  and  (6),  the  result  of  eliminating  the  unknowns  is 
independent  of  the  order  in  which  they  are  eliminated. 
Since  this  theorem  holds,  it  will  be  unnecessary  to  investigate  separately 

the  equation  of  condition  for  the  various  forms  of  equations;    they  are 

already  given  in  the  theorems  concerning  elimination. 

7-51  Any  equation  in  n  unknowns,  of  the  form  F(XI,  x2,  ...  xn)  =  0, 
provided  its  equation  of  condition  be  satisfied,  gives  a  solution  for  each 
of  the  unknowns  as  follows:  Let  Xk  be  any  one  of  the  unknowns;  let  PI,  PZ, 
P3,  etc.,  be  the  coefficients  of  those  terms  in  F(XI,  xz>  ...  xn)  in  which  Xk 
is  positive,  and  Qi,  Qz,  Qs,  etc.,  the  coefficients  of  those  terms  in  which  Xk 
is  negative.  The  solution  then  is 

JJ  Q  c  xk  c  ^  -P,     or    Xk  =  II  Q  +  u  •  ^  -P,  where  u  is  arbitrary. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  157 

(a)  By  6-11,  for  some  /  and  some  /',  F(x\.,  x2,  ...  xn)  =  0  is 
equivalent  to  f(xi,  x2,  ...  xn-i)  -xn+f  '(xi,  x2,  ...  xn-i}--xn  =  0. 
Let  the  coefficients  in  /  be  PI,  P2,  PS,  etc.,  in  /  '  be  Qi,  Q2,  Qs,  etc. 
Then  PI,  P2,  PS,  etc.,  will  be  the  coefficients  of  those  terms  in  F 
in  which  Xk  is  positive,  Qi,  Q2,  Qs,  the  coefficients  of  terms  in  F  in 
which  Xk  is  negative. 

If  f(xi,  xz,  ...  xn-i)-xn  be  regarded  as  a  function  of  the  variables, 
xi}  Xz,  ...  xn-i,  its  coefficients  will  be  P\xn,  P2xn,  P3xn,  etc. 
And  if  f'(xi,  x2,  ...  xn-i)--xn  be  regarded  as  a  function  of  Xi,  x2, 
.  .  .  xn-i,  its  coefficients  will  be  Qi  -xn,  Q2  -xn,  Qs  -xn,  etc. 
Hence,  by  6-42, 

f(Xi,  XZ,    .  .  .    Xn-j_)  -Xn+f  '(Xi,  X2)    ...    Xr^i)"Xn   =   0 


is  equivalent  to  ^(xi,  x2,  .  .  .  xn-\)  =  0,  where  ^  is  a  function  in 
which  the  coefficients  are  (Pi  xn  +  Qi  -xn},  (Pz  xn+  Q2  -xn),  (P3xn 
+  Qs  -Xn),  etc. 

And  SlK^i,  xz,  ...  £n-i)  =  0  is  equivalent  to  F(XI,  x2,  .  .  .  xn)  =  0. 
By  7-41,  the  complete  resultant  of  the  elimination  of  Xi,  x2,  ...  xn-i 
from  ^(#1,  Xz,  ...  Xn-i)  =  0  will  be  the  equation  to  0  of  the  product 
of  its  coefficients,— 

Qr-xJ  =0 


But  any  expression  of  the  form  PrXn  +  QT  -xn  is  a  normal  form  func- 
tion of  xn.     Hence,  by  6  •  43, 

II  (P#n  +  Qr  -Xn)    =   H    P#n  +    II  Qr  ~Xn 


By  7-2  and  7-24,  the  solution  of  H  P^n+  TlQr  -xn  =  0  i 


s 


,     or    xn  =  +  u. 

And  [5-951]  -{UP}  =  Z-J°- 

(6)  Since  the  order  of  terms  in  a  function  is  immaterial,  and 
for  any  order  of  the  variables  in  the  argument  of  a  function  there  is  a 
normal  form  of  the  function,  xn  in  the  above  may  be  any  one  of  the 
variables  in  F(XI,  x2,  ...  xn),  and  f(xi,  x2,  ...  xn-J  and  f'(xi,  x2, 
.  .  .  xn-i)  each  some  function  of  the  remaining  n  —  1  variables. 
Therefore,  the  theorem  holds  for  any  one  of  the  variables,  Xk- 
That  a  single  equation  gives  a  solution  for  any  number  of  unknowns 

is  another  peculiarity  of  the  algebra,  due  to  the  fact  that  from  a  single 

equation  any  number  of  unknowns  may  be  eliminated. 


158  A  Survey  of  Symbolic  Logic 

As  an  example  of  the  last  theorem,  we  give  the  solution  of  the  exemplar 
equation  in  two  unknowns,  first  directly  from  the  theorem,  then  by  elimina- 
tion and  solution  for  each  unknown  separately. 

(1)  A  x  y  +  B  -x  y  +  C  x  -y  +  D  -x  -y  =  0  has  the  equation  of  condition, 

ABCD  =  0 
Provided  this  be  satisfied,  the  solutions  for  x  and  y  are 

B  DC.  re  -A  +-C,        or        x  =  B  D  +  u  (-A  +  -C) 
CDcyc-A+-B,        or        y  =  C  D  +  u  (-A  +  -B) 

(2)  A  x  y  +  B  -x  y  +  C  x  -y  +  D  -x  -y  =  0  is  equivalent  to 

(a)     (A  x  +  B  -«)  y  -f  (C  x  +  D  -x)  -y  =  0 
and  to  (6)     (A  y  +  C  -y}  x  +  (B  y  +  D  -y}  -x  =  0 
Eliminating  y  from  (a),  we  have 


The  equation  of  condition  with  respect  to  x  is,  then, 
(AC)(BD)  =  ABCD  =  0 
And  the  solution  for  x  is 

B  D  ex  c-(A  C),     or    x  =  B  D  +  ir—(A  C).     And  -(.4  C)  =  -A  +  -C 
Eliminating  x  from  (6),  we  have 

(Ay+C-y)(By  +  D-y)  =  ABy+CD-y  =  0 

The  equation  of  condition  with  respect  to  y  is,  then,  ABCD  =  0.     And 
the  solution  for  y  is 

CDcyc-(AB),    or    y  =  CD+v-(AB).     And  -(A  B}  =  -A  +  -B 

Another  method  of  solution  for  equations  in  two  unknowns,  x  and  y, 
would  be  to  solve  for  y  and  for  -y  in  terms  of  the  coefficients,  with  x  and  u 
as  undetermined  parameters,  then  eliminate  y  by  substituting  this  value 
of  it  in  the  original  equation,  and  solve  for  x.  By  a  similar  substitution, 
x  may  then  be  eliminated  and  the  resulting  equation  solved  for  y.  This 
method  may  inspire  more  confidence  on  the  part  of  those  unfamiliar  with 
this  algebra,  since  it  is  a  general  algebraic  method,  except  that  in  other 
algebras  more  than  one  equation  is  required. 

The  solution  of  A  x  y  +  B  -x  y+  C  x  -y  +  D  -x  -y  =  0  for  y  is 

y  =  (C  x  +  D  -x)  +  u--(A  x  +  B  -x}  =  (C  +  u  -A)  x  +  (D  +  u  -B)  -x 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  159 

The  solution  for  -y  is 

-y  =  (Ax  +  B  -x}  +  v-(C  x  +  D -x)  =  (A+v  -C)  x  +  (B  +  v  -D}  -x 
Substituting  these  values  for  y  and  -y  in  the  original  equation, 

(Ax  +  B  -x)[(C  +  u-A)x  +  (D  +  u  -B)  -x] 

+  (C  x  +  D  -x)[(A  +  v-C)x  +  (B  +  v  -D)  -x\ 

=  A  (C  +  u  -A]  x  +  B  (D  +  u  -B}  -x  +  C  (A  +  v  -C)  x  +  D  (B  +  v  -D)  -x 

=  ACx  +  BD-x  =  0. 
Hence  BDcxc-A+-C. 

Theoretically,  this  method  can  be  extended  to  equations  in  any  number  of 
unknowns:  practically,  it  is  too  cumbersome  and  tedious  to  be  used  at  all. 

7  •  52     Any  equation  in  n  unknowns,  of  the  form 

F(XI,  x2,  ...  xn)  =  f(xi,  xz,  ...  xn) 

gives  a  solution  for  each  of  the  unknowns  as  follows:  Let  xk  be  any  one 
of  the  unknowns;  let  PI,  P2,  PS,  •  •  •  Qi,  Qz,  Qs,  •  •  •  be  the  coefficients  in  F, 
and  MI,  MI,  Ms,  .  .  .  NI,  NZ,  N3,  .  .  .  the  coefficients  of  the  corresponding 
terms  in  /,  so  that  Pr  and  Mr  are  coefficients  of  terms  in  which  xk  is  positive, 
and  Qr  and  Nr  are  coefficients  of  terms  in  which  Xk  is  negative.  The  solu- 
tion for  Xk  then  is 

II  (Qr  -Nr  +  -Qr  N r)  c  xk  c  £  (Pr  Mr  +  -Pr  -MJ 

r  r 

Or  Xk    =   II    (Qr  -Nr  +  -Qr  Nr)+U-^   (Pr  Mr  +  -Pr  ~Mr) 

r  r 

By   7-13,    F(XI,  x2,  ...  Xn)  =  f(xi,  x2,  ...  xn)    is   equivalent   to 
<b(xi,  Xz,  ...  xn)  =  0,  where  4>  is  a  function  such  that  if  Ar  and  Br 
be  coefficients  of  any  two  corresponding  terms  in  F  and  /,  then  the 
coefficient  of  the  corresponding  term  in  $  will  be  Ar  -Br  +  -Ar  Br. 
Hence,  by  7-51,  the  solution  will  be 

II     (Q,  -Nr  +  -Qr  Nr~)   C  Xk  C   £  -(Pr  ~Mr  +  ~P r  M r) 
r  r 

Or  Xk    =   II    (Qr  -Nr  +  -Qr  N r)  +  U  •  ^  "(Pr  "Mr  +  -Pr  Jf  r) 

r  r 

And  [6-4]  -(Pr-i¥r  +  -Pri¥r)  =  (Pr  Mr  +  -Pr  -  M  r) . 

7-53  The  condition  that  m  equations  in  n  unknowns,  each  of  the  form 
F(XI,  Xz,  ...  xn)  =  0,  may  be  regarded  as  simultaneous,  is  as  follows: 
Let  the  coefficients  of  the  terms  in  F1,  in  the  equation  Fl(xi}  x2,  ...  xn}  =  0, 
be  P^,  P2S  Pg1,  .  .  .  Q!1,  Q2J,  Qsl,  ...;  let  the  coefficients  of  the  corre- 


160  A  Survey  of  Symbolic  Logic 

spending  terms  in  F2,  in  the  equation  F2(xi,  a*2,  ...  xn)  =  0,  be  Pi2,  P22, 
PS,  •  •  •  Qi2,  Qz2,  Qs2,  •  •  .',  the  coefficients  of  the  corresponding  terms  in 
Fm,  in  the  equation  Fm(xi,  x2,  ...  xn)  =  0,  be  Pi"1,  P2m,  P3m,  .  .  .  Qim,  Q2m, 
Q3m,  ....  The  condition  then  is 


r         A  r         h 

Or  if  C/  be  any  coefficient,  whether  P  or  Q,  in  PA,  the  condition  is 


r          A 

And  the  solution  which  n  such  equations  give,  on  this  condition,  for  any 
one  of  the  unknowns,  xk,  is  as  follows:  Let  PA  P2A,  Pzh,  ...  be  the  coef- 
ficients of  those  terms,  in  any  one  of  the  equations  Fh  =  0,  in  which  Xk  is 
positive,  and  let  Qih,  Q2h,  Q,3h,  ...  be  the  coefficients  of  those  terms,  in 
Fh  =  0,  in  which  a*&  is  negative.  The  solution  then  is 


or 

r          h  r          h 

By  6-42,  m  equations  in  n  unknowns,  each  of  the  form  F(x\,  z2, 
. .  .  xn)  =  0,  are  together  equivalent  to  the  single  equation  $(xi,  x2, 
...£„)=  0,  where  each  of  the  coefficients  in  $  is  the  sum  of  the 
corresponding  coefficients  in  F1,  F2,  F3,  ...  Fm.  That  is,  if  Prl,  Pr2, 
.  . .  Prm  be  the  coefficients  of  corresponding  terms  in  F1,  F2,  ...  Fm, 
then  the  coefficient  of  the  corresponding  term  in  <£  will  be 

Pr1  +  Pr2  +   .  .   .    +  Pr>»,  Or  X)  Prk 

A 

and  if  Qrl,  Qr2,  .  .  .  Qrm  be  the  coefficients  of  corresponding  terms  in 
F1,  F2,  ...  Fm,  then  the  coefficient  of  the  corresponding  term  in  $ 
will  be 


The  equation  of  condition  for  $  =  0,  and  hence  the  condition  that 
F1  =  0,  F2  =  0,  ...  Fm  =  0  may  be  regarded  as  simultaneous,  is 
the  equation  to  0  of  the  product  of  the  coefficients  in  <£;  that  is, 

2  P!*  X  £  P2"  X  £  P3"  X  .  .  .    X  2  <3l"  ><  Z  QS  ><  Z  #3*  X  .  .  .     =0 

A  A  A  A  A  A 


or 

r          h 


And  by  7-51,  the  solution  of  $(0:1,  £2,  .  .  .  .rn)  =  0  for  Xk  is 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  161 


or  afc= 

r          h  r  h 


And  by  5-95,  -[£  PP»]  =        -PA 

/i  A 

7  •  54     The  condition  that  m  equations  in  n  unknowns,  each  of  the  form 
F(XI,  x2,  ...  xn)  =  f(xi,  x2,  ...  Xn) 

may  be  regarded  as  simultaneous,  is  as  follows:  Let  the  coefficients  in  F1, 
in  the  equation  F1  =fl,  be  Pi1,  Pa1,  Ps1,  .  .  .  Qi1,  &1,  Qa1,  .  .  .,  and  let  the 
coefficients  of  the  corresponding  terms  in  /  l,  in  the  equation  F1  =  fl,  be 
Mi1,  Mzl,  M3l,  .  .  .  Ni1,  N2l,  Ns1,  .  .  .;  let  the  coefficients  of  the  corresponding 
terms  in  F2,  in  the  equation  F2  =  f2,  be  P^,  P22,  P32,  .  .  .  Qi2,  Q22,  Q32, 
and  let  the  coefficients  of  the  corresponding  terms  in  /  2  be  M  i2,  M  22,  M  32, 
.  .  .  Ni2,  Nzz,  Ns2,  .  .  .  ;  let  the  coefficients  of  the  corresponding  terms  in  Fm, 
in  the  equation  Fm  =  /  m,  be  Pim,  P2m,  P3TO,  .  .  .  Qim,  Qzm,  Qsm,  ••-,  and 
let  the  coefficients  of  the  corresponding  terms  in  /  m  be  Mim,  Mzm,  Msm, 
.  .  .  Nim,  N2m,  N3m,  ....  The  condition  then  is 

II    I  E   (P/  ~Mrh  +  -Prh  Mrh}}   X   H  [  E  (&*  -#  r*  +  -Qr*  ^r*)]    =    0 
r          h  r          h 

or  if  Arh  represent  any  coefficient  in  Fh,  whether  P  or  Q,  and  Brh  represent 
the  corresponding  coefficient  in  /  h,  whether  M  or  N,  the  condition  is 

II    [E(^r*-Br*  +  -^r*5r*)]    =0 

r          h 

And  the  solution  which  m  such  equations  give,  on  this  condition,  for  any 
one  of  the  unknowns,  Xk,  is  as  follows:  Let  Prh  and  Mrh  be  the  coefficients 
of  those  terms,  in  any  one  of  the  equations  Fh  =  fh,  in  which  xk  is  positive, 
and  let  Qrh  and  Nrh  be  the  coefficients  of  the  terms,  in  Fh  =  f  h,  in  which  Xk 
is  negative.  The  solution  then  is 

II    [  E   (Qrk  -Nrk  +  ~Qrh  N*)]  C  Xk  C   E  til    (P^  Mrh  +  ~P  rh  ~MTh}} 
r          h  r          h 

Or       Xk    =   I 

r 

By  7-13,  .FA(£i,  X2,  ...  £„)  =  /A(a;:,  xz,  ...  ^n)  is  equivalent  to 
ty(xi,  x2,  ...  a;n)  =  0,  where  ^  is  a  function  such  that  if  Qrh  and  Nrh 
be  coefficients  of  corresponding  terms  in  Fh  and  /  h,  the  coefficient 
of  the  corresponding  term  in  ^  will  be  Qrh  -Nrh  +  -Qrh  Nrh,  and  if 
Pr*  and  Mrh  be  coefficients  of  corresponding  terms  in  Fh  and  /  h,  the 
coefficient  of  the  corresponding  term  in  ^  will  be  Pr*  -Mrh  +  -Prh  Mrh. 
And  -(Prh  -Mrh  +  -Prh  Mrh)  =  Prh  Mrh  +  -Pr*  -3fr\ 
Hence  the  theorem  follows  from  7  •  53. 
12 


162  A  Survey  of  Symbolic  Logic 

F(XI,  x2,  . . .  Xn)  =  f(x\,  #2,  ...  xn}  is  a  perfectly  general  equation,  since 
F  and  /  may  be  any  expressions  in  the  algebra,  developed  as  functions  of 
the  variables  in  question.  7-54  gives,  then,  the  condition  and  the  solution 
of  any  number  of  simultaneous  equations,  in  any  number  of  unknowns,  for 
each  of  the  unknowns.  This  algebra  particularly  lends  itself  to  generaliza- 
tion, and  this  is  its  most  general  theorem.  It  is  the  most  general  theorem 
concerning  solutions  in  the  whole  of  mathematics. 

Boole's  General  Problem. — Boole  proposed  the  following  as  the  general 
problem  of  the  algebra  of  logic.11 

Given  any  equation  connecting  the  symbols  x,  y,  ...  w,  z,  ....  Re- 
quired to  determine  the  logical  expression  of  any  class  expressed  in  any 
way  by  the  symbols  x,  y,  ...  in  terms  of  the  remaining  symbols  w,  z,  .... 
We  may  express  this:  Given  t  =  f(x,  y,  .  .  .)  and  <£(£,  y,  . .  . )  =  ^(w,  z, 
. . . ) ;  to  determine  t  in  terms  of  w,  z,  ....  This  is  perfectly  general,  since 
if  x,  y,  ...  and  w,  z,  ...  are  connected  by  any  number  of  equations,  there 
is,  by  7-1  and  5-72,  a  single  equation  equivalent  to  them  all.  The  rule 
for  solution  may  be  stated:  Reduce  both  t  =  f(x,  y,  .  .  .)  and  $(x,  y,  .  .  .) 
=  ^ (w,  z,  . . . )  to  the  form  of  equations  with  one  member  0,  combine  them 
by  addition  into  a  single  equation,  eliminate  x,  y,  .  .  . ,  and  solve  for  t.  By 
7-1,  the  form  of  equation  with  one  member  0  is  equivalent  to  the  other 
form.  And  by  5-72,  the  sum  of  two  equations  with  one  member  0  is 
equivalent  to  the  equations  added.  Hence  the  single  equation  resulting 
from  the  process  prescribed  by  our  rule  will  contain  all  the  data.  The 
result  of  eliminating  will  be  the  complete  resultant  which  is  independent 
of  these,  and  the  solution  for  t  will  thus  be  the  most  complete  determination 
of  t  in  terms  of  w,  z,  ...  afforded  by  the  data. 

Consequences  of  Equations  in  General. — A  word  of  caution  with  refer- 
ence to  the  manipulation  of  equations  in  this  algebra  may  not  be  out  of 
place.  As  compared  with  other  algebras,  the  algebra  of  logic  gives  more 
room  for  choice  in  this  matter.  Further,  in  the  most  useful  applications 
of  the  algebra,  there  are  frequently  problems  of  procedure  which  are  not 
resolved  simply  by  eliminating  this  and  solving  for  that.  The  choice  of 
method  must,  then,  be  determined  with  reference  to  the  end  in  view.  But 
the  following  general  rules  are  of  service: 

(1)  Get  the  completest  possible  expression  =  0,  or  the  least  inclusive 
possible  expression  =  1. 
a  +  b  +  c+  .  .  .  =0      gives      a  =  0,  6  =  0,  c  =  0,  .  . . ,  a  +  6  =  0,  a  +  c  =  0, 

"Laws  of  Thought,  p.  140. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  163 

etc.  But  a  =  0  will  not  generally  give  a  +  6  =  0,  etc.  Also,  a  =  1  gives 
a  +  b  =  1,  a+  .  .  .  =  1,  but  a  +  b  =  1  will  not  generally  give  a  =  1. 

(2)  Reduce  any  number  of  equations,  with  which  it  is  necessary  to  deal, 
to  a  single  equivalent  equation,  by  first  reducing  each  to  the  form  in  which 
one  member  is  0  and  then  adding.  The  various  constituent  equations 
can  always  be  recovered  if  that  be  desirable,  and  the  single  equation  gives 
other  derivatives  also,  besides  being  easier  to  manipulate.  Do  not  forget 
that  it  is  possible  so  to  combine  equations  that  the  result  is  less  general 
than  the  data.  If  wTe  have  a  =  0  and  6  =  0,  we  have  also  a  —  b,  or  a  b  =  0, 
or  a  +  6  =  0,  according  to  the  mode  of  combination.  But  a  +  b  =  0  is 
equivalent  to  the  data,  while  the  other  two  are  less  comprehensive. 

A  general  method  by  which  consequences  of  a  given  equation,  in  any 
desired  terms,  may  be  derived,  was  formulated  by  Poretsky,12  and  is,  in 
fact,  a  corollary  of  his  Law  of  Forms,  given  above.  We  have  seen  that 
this  law  may  be  formulated  as  the  principle  that  if  a  =  b,  and  therefore 
a-b  +  -ab  =  0  and  ab  +  -a-b  =  1,  then  any  t  is  such  that  a  -b  +  -a  b  c  t 
and  tcab  +  -a-b,  or  any  t  =  the  t  which  contains  the  "zero  member" 
of  the  set  of  equations  equivalent  to  a  —  b,  and  is  contained  in  the  "  whole 
member"  of  this  set.  Now  if  x  c  t,  u  x  c  t,  for  any  u  whatever,  and  thus  the 
"zero  member"  of  the  Law  of  Forms  may  be  multiplied  by  any  arbitrarily 
chosen  u  which  we  choose  to  introduce.  Similarly,  if  t  c  y,  then  t  c  y  +  v, 
and  the  "whole  member"  in  the  Law  of  Forms  may  be  increased  by  the 
addition  of  any  arbitrarily  chosen  v.  This  gives  the  Law  of  Consequences. 
7-6  If  a  =  b,  then  t  =  (a  b  +  -a  -b  +  u)  t  +  v  (a  -b  +  -a  b)  -t,  where  u  and  v 
are  arbitrary. 

[7  •  1  •  12J     If  a  =  b,  then  a  -b  +  -a  b  =  0  and  ab  +  -a-b  =  1. 
Hence  (a  b  +  -a  -b  +  u)  t  +  v  (a  -b  +  -a  6)  -t  =  (1  +  u)  t  +  v  •  0  •  -i  --=  t. 

This  law  includes  all  the  possible  consequences  of  the  given  equation. 
First,  let  us  see  that  it  is  more  general  than  the  previous  formulae  of  elimina- 
tion and  solution.  Given  the  equation  A  x  +  B  -x  =  0,  and  choosing  A  B 
for  t,  we  should  get  the  elimination  resultant. 

If  A  x  +  B  -x  =  0,  then  A  B  =  (-A  x  +  -B  -x  +  u}  A  B 

+  v(Ax  +  B-x)(-A+-B) 
=  u  A  B  +  v  (A  -B  x  +  -A  B  -x). 

Since  u  and  v  are  both  arbitrary  and  may  assume  the  value  0,  there- 
fore AB  =  0. 
12  Sept  lois,  etc.,  Chap.  xn. 


164  A  Survey  of  Symbolic  Logic 

But  this  is  only  one  of  the  unlimited  expressions  for  A  B  which  the  law 
gives.     Letting  u  =  0,  and  v  =  1,  we  have 

A  B  =  A  -B  x  +  -A  B  -x 
Letting  u  =  A  and  v  =  B,  we  have 

AB  =  AB  +  -A  B  -x 

And  so  on.     But  it  will  be  found  that  every  one  of  the  equivalents  of  A  B 

which  the  law  gives  will  be  null. 

Choosing  x  for  our  t,  we  should  get  the  solution. 

If  A  x  +  B  -x  =  0,  then  x  =  (-A  x  +  -B  -x  +  ii)  x  +  v  (A  x  +  B  -x)  -x 

=  (-A  +  w)  x  +  v  B  -x. 

Since  u  and  v  may  both  assume  the  value  0, 

x  =  -A  x,        or        x  c  -A  (1) 

And  since  u  and  v  may  both  assume  the  value  1, 
x  =  x  +  B  -x,        or        B  -x  c  x 

But  if  £  -a;  c  x,  then  B  -x  =  (B  -x)  x  =  0,  or  B  c  x  (2) 

Hence,  (1)  and  (2),  Bex  c-A. 

When  u  =  0  and  v  =  1,  the  Law  of  Consequences  becomes  simply  the  Law 
of  Forms.  For  these  values  in  the  above, 

x  =  -A  x  +  B  -x 

which  is  the  form  which  Poretsky  gives  the  solution  for  x. 

The  introduction  of  the  arbitraries,  u  and  v,  in  the  Law  of  Consequences 
extends  the  principle  stated  by  the  Law  of  Forms  so  that  it  covers  not 
only  all  equivalents  of  the  given  equation  but  also  all  the  non-equivalent 
inferences.  As  the  explanation  which  precedes  the  proof  suggests,  this  is 
accomplished  by  allowing  the  limits  of  the  function  equated  to  t  to  be 
expressed  in  all  possible  ways.  If  a  =  b,  and  therefore,  by  the  Law  of 
Forms, 

t  =  (a  b  +  -a  -6)  t  +  (a  -b  +  -a  6)  -t 

the  lower  limit  of  t,  0,  is  expressed  as  a  -b  +  -a  b,  and  the  upper  limit  of  t, 
1,  is  expressed  as  a  b  +  -a  -b.  In  the  Law  of  Consequences,  the  lower 
limit,  0,  is  expressed  as  v  (a-b  +  -ab),  that  is,  in  all  possible  wrays  which 
can  be  derived  from  its  expression  as  a-b  +  -ab;  and  the  upper  limit,  1,  is 
expressed  as  a  b  +  -a  -b  +  u,  that  is,  in  all  possible  ways  which  can  be  derived 
from  its  expression  as  a  b  +  -a  -6.  Since  an  expression  of  the  form 

t  =  (a  b  +  -a  -b)  t  +  (a  -b  +  -a  b)  -t 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  165 

or  of  the  form  t  =  (a  b  +  -a  -b  +  w)  t  +  v  (a  -b  +  -a  6)  -t 
determines  t  only  in  the  sense  of  thus  expressing  its  limits,  and  the  Law  of 
Consequences  covers  all  possible  ways  of  expressing  these  limits,  it  covers 
all  possible  inferences  from  the  given  equation.  The  number  of  such 
inferences  is,  of  course,  unlimited.  The  number  expressible  in  terms  of  n 
elements  will  be  the  number  of  derivatives  from  an  equation  with  one 
member  0  and  the  other  member  expanded  with  reference  to  n  elements. 
The  number  of  constituent  terms  of  this  expanded  member  will  be  2n, 
and  the  number  of  combinations  formed  from  them  will  be  22?l.  Therefore, 
since  p\  +  p2  +  PS  +  . . .  =0  gives  pi  =  0,  p2  =  0,  p3  =  0,  etc.,  this  is  the 
number  of  consequences  of  a  given  equation  which  are  expressible  in  terms 
of  n  elements. 

As  one  illustration  of  this  law,  Poretsky  gives  the  sixteen  determinations 
of  a  in  terms  of  the  three  elements,  a,  b,  and  c,  which  can  be  derived  from 
the  premises  of  the  syllogism  in  Barbara: 13 

If  all  a  is  b,         a  -b  —  0, 
and  all  b  is  c,         b  -c  =  0, 
then  a-b  +  b  -c  =  0,  and  hence, 

a  =  a  (b  +  -c)  =  a  (b  +  c)  =  a  (-b  +  c)  =  a  +  b  -c  =  a  b  =  a  (b  c  +  -b  -c) 
=  b  -c  +  a  (b  c  +  -b  -c)  =ac  =  b-c  +  ac  =  a  (-6  +  c)  +  -a  b  -c  =  ab  c 
=  b  -c  +  ab  c  =  a  (b  c  +  -b  -c)  +  -a  b  -c  =  a  c  +  -a  b  -c  =  a  b  c  +  -a  b  -c 

The  Inverse  Problem  of  Consequences. — Just  as  the  Law  of  Conse- 
quences expresses  any  inference  from  a  =  b  by  taking  advantage  of  the  fact 
that  if  a-b  +  -ab  =  0,  then  (a-b  +  -ab)v  =  0,  and  if  ab  +  -a-b  =  1, 
then  a  b  +  -a  -b  +  u  =  1 ;  so  the  formula  for  any  equation  which  will  give 
the  inference  a  =  b  can  be  expressed  by  taking  advantage  of  the  fact  that  if 
v  (a  b  +  -a  -b)  =  1,  then  ab  +  -a-b  =  1,  and  if  a  -b  +  -a  b  +  u  =  0,  then 
a-b  +  -ab  =  0.  We  thus  get  Poretsky's  Law  of  Causes,  or  as  it  would 
be  better  translated,  the  Law  of  Sufficient  Conditions.14 
7  •  7  If  for  some  value  of  u  and  some  value  of  v 

t  =  v  (a  b  +  -a  -6)  t  +  (a  -b  +  -a  b  +  u)  -t, 
then  a  =  b. 

If  t  =  v  (a  b  +  -a  -6)  t  +  (a  -b  +  -a  b  +  u)  -t,  then  [7-1,  5-72] 
[v  (a  b  +  -a  -b)  t  +  (a  -b  +  -a  b  +  it)  -t]  -t  =  0 

=  (a  -b  +  -a  b  +  u)  -t  =  (a  -b  +  -a  6)  -t  +  u  -t  =  0 

13  Ibid.,  pp.  QBff. 

14  Ibid.,  Chap.  xxm. 


166  A  Survey  of  Symbolic  Logic 

Hence  (a  -b  +  -a  b)  -t  =  0  (1) 

Hence  also  [5-7]  t  =  v  (ab  +  -a-b)  t,  and  [4-9] 

t  •  -[v  (a  b  +  -a  -b)]  =  0  =  t  (-v  +  a  -b  +  -a  6)  =  t  -v  +  (a  -6  +  -a  6)  t 

Hence  [5  •  72]  (a  -6  +  -a  6)  t  =  0  (2) 

By  (1)  and  (2),  (a  -6  +  -a  6) (t  +  -t)  =  0  =  a  -6  +  -a  b. 
Hence  [7-1]  a  =  b. 

Both  the  Law  of  Consequences  and  the  Law  of  Sufficient  Conditions 
are  more  general  than  the  Law  of  Forms,  which  may  be  derived  from  either. 
Important  as  are  these  contributions  of  Poretsky,  the  student  must  not 
be  misled  into  supposing  that  by  their  use  any  desired  consequence  or 
sufficient  condition  of  a  given  equation  can  be  found  automatically.  The 
only  sense  in  which  these  laws  give  results  automatically  is  the  sense  in 
which  they  make  it  possible  to  exhaust  the  list  of  consequences  or  conditions 
expressible  in  terms  of  a  given  set  of  elements.  And  since  this  process  is 
ordinarily  too  lengthy  for  practical  purposes,  these  laws  are  of  assistance 
principally  for  testing  results  suggested  by  some  subsidiary  method  or  by 
"intuition  ".  One  has  to  discover  for  himself  what  values  of  the  arbitraries 
u  and  v  will  give  the  desired  result. 

V.    FUNDAMENTAL  LAWS  OF  THE  THEORY  OF  INEQUATIONS 

In  this  algebra,  the  assertory  or  copulative  relations  are  =  and  c . 
The  denial  of  a  =  b  may  conveniently  be  symbolized  in  the  customary  way: 

8-01  a  3=  b  is  equivalent  to  "a  =  b  is  false  ".  Def. 
We  might  use  a  symbol  also  for  "a  c  b  is  false  ".  But  since  a  c  b  is  equiva- 
lent to  a  b  =  a  and  to  a  -b  =  0,  its  negative  may  be  represented  by  a  b  4=  a 
or  by  a  -6  4=  0.  It  is  less  necessary  to  have  a  separate  symbolism  for 
"acb  is  false  ",  since  "a  is  not  contained  in  b"  is  seldom  met  with  in  logic 
except  where  a  and  b  are  mutually  exclusive,— in  which  case  a  b  —  0. 

For  every  proposition  of  the  form  "If  P  is  true,  then  Q  is  true  ",  there  is 
another,  "  If  Q  is  false,  then  P  is  false  ".  This  is  the  principle  of  the  reductio 
ad  absurdum, — or  the  simplest  form  of  it.  In  terms  of  the  relations  = 
and  4=>  the  more  important  forms  of  this  principle  are: 

(1)  "If  a  =  b,  then  c  =  d",  gives  also,  "If  c  4=  d,  then  a  4=  6  ". 

(2)  "If  a  =  b,  then  c  =  dandh  =  k",  gives  also,  "  If  c  4=  Athena  =|=  6", 
and  "If  h  =1=  Mhena  *  b". 

(3)  "If  a  =  b  and  c  =  d,  then  h  =  k",  gives  also,  "If  a  =  b  and  h  =|=  k, 
then  c  4=  d",  and  "  If  c  =  d  and  h  =(=  A-,  then  a  4=  b  ". 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  167 

(4)  "a  =  b  is  equivalent  to  c  =  d",  gives  also,  "a  4=  b  is  equivalent 
toe  4=  d". 

(5)  "a  =  b  is  equivalent  to  the  set,  c  =  d,  h  =  k,    .  .  .,"  gives  also, 
"a  =}=  b  is  equivalent  to  'Either  c  =j=  ^  or  A  =|=  k,  or  ...'".  16 

The  general  forms  of  these  principles  are  themselves  theorems  of  the 
"calculus  of  propositions" — the  application  of  this  algebra  to  propositions. 
But  the  calculus  of  propositions,  as  an  applied  logic,  cannot  be  derived 
from  this  algebra  without  a  circle  in  the  proof,  for  the  reasoning  in  demon- 
stration of  the  theorems  presupposes  the  logical  laws  of  propositions  at 
every  step.  We  must,  then,  regard  these  laws  of  the  reductio  ad  absurdum, 
like  the  principles  of  proof  previously  used,  as  given  us  by  ordinary  logic, 
which  mathematics  generally  presupposes.  In  later  chapters,16  we  shall 
discuss  another  mode  of  developing  mathematical  logic — the  logistic 
method — wrhich  avoids  the  paradox  of  assuming  the  principles  of  logic  in 
order  to  prove  them.  For  the  present,  our  procedure  may  be  viewed  simply 
as  an  application  of  the  reductio  ad  absurdum  in  ways  in  which  any  mathe- 
matician feels  free  to  make  use  of  that  principle. 

Since  the  propositions  concerning  inequations  follow  immediately,  for 
the  most  part,  from  those  concerning  equations,  proof  will  ordinarily  be 
unnecessary. 

Elementary  Theorems. — The  more  important  of  the  elementary  propo- 
sitions are  as  follows: 

8-1     If  a  c  =%=  b  c,  then  a  4=  b. 

[2-1] 
8-12     If  a  +  c  4=  b  +  c,  then  a  4=  b. 

[3-37] 
8-13     a  =1=  b  is  equivalent  to  -a  =)=  -b. 

[3-2] 
8-14     a  +  b  4=  b,  a  b  =4=  a,  -a  +  b  =f=  1,  and  a  -b  4=  0  are  all  equivalent. 

[4-9] 
8-15     If  a  +  b  =  x  and  b  =f=  %,  then  a  4=  0 

[5-7] 
8-151     If  a  =  0  and  b  4=  x,  then  a  +  b  4=  %• 

[5-7] 
8-16     If  a  b  =  x  and  b  3=  x,  then  a  4=  1. 

[5-71] 

15  "Either  ...  or  ..."  is  here  to  be  interpreted  as  not  excluding  the  possibility  that 
both  should  be  true. 

18  Chap,  iv,  Sect,  vi,  and  Chap.  v. 


168  A  Survey  of  Symbolic  Logic 

8- 161     If  a  =  1  and  6  4=  x,  then  ab  =£  x. 

[5-71] 

8-17     If  a  +  b  4=  0  and  a  =  0,  then  b  ={=  0. 

[5-72] 
8-18     If  a  6  4=  1  and  a  =  1,  then  6  4=  1. 

[5-73] 

8-17  allows  us  to  drop  null  terms  from  any  sum  4=  0.  In  this,  it  gives 
a  rule  by  which  an  equation  and  an  inequation  may  be  combined.  Suppose, 
for  example,  a  +  b  4=  0  and  x  =  0. 

a  +  b  =  (a  +  6)  (x  +  -x)  =  ax+bx+a-x+b  -x. 
Hence  a  x  +  b  x  +  a  -x  +  b  -x  4=  0. 
But  if  x  =  0,  then  a  x  =  0  and  b  x  =  0. 
Hence  [8  •  17]  a  -x  +  b  -x  4=  0. 

8-2     If  a  4=  0,  then  a  +  b  4=  0. 

[5-72] 
8-21     If  a  4=  1,  then  ab  4=  1. 

[5-73] 
8-22     If  a  b  4=  0,  then  a  4=  0  and  6  4=  0. 

[1-5] 
8-23     If  a+b  4=  1,  then  a  4=  1  and  b  4=  1. 

[4-5] 
8-24     If  a  6  4=  #  and  a  =  x,  then  b  =]=  a:. 

[1-2] 
8-25     If  a  4=  0  and  a  c  6,  then  6  =1=  0. 

[1-9]  If  a  c  6,  then  a  b  =  a. 
Hence  if  a  4=  0  and  a  c  6,  then  a  6  4=  0. 
Hence  [8-22]  6  +  0. 

8-26     a  +  6  4=  0  is  equivalent  to  "Either  a  4=  0  or  b  4=  0  ". 

[5-72] 

8-261     di  +  az  +  a3+  . . .  4=  0  is  equivalent  to  "Either  ax  4=  0  or  a2  4=  0  or 
o8  4=  0,  or  . .  .  ". 
8-27     a  6  4=  1  is  equivalent  to  "Either  a  4=  1  or  6  4=  1  ". 

[5-73] 

8-271     ai  a2  «3  . . .  4=  1    is   equivalent   to   "Either  ax  4=  1    or   a2  4=  1    or 
a,  4=  1  or  . . .  ". 

The  difference  between  8-26  and  8-27  and  their  analogues  for  equa- 
tions— 5  -72  a  +  b  —  0  is  equivalent  to  the  pair,  a  =  0  and  6  =  0,  and 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  169 

5-73  a  b  =  1  is  equivalent  to  the  pair,  a  =  1  and  b  =  1  —  points  to  a  neces- 
sary difference  between  the  treatment  of  equations  and  the  treatment  of 
inequations.  Two  or  more  equations  may  always  be  combined  into  an 
equivalent  equation;  two  or  more  inequations  cannot  be  combined  into 
an  equivalent  inequation.  But,  by  8-2,  a+  b  =(=  0  is  a  consequence  of  the 
pair,  a  =f=  0  and  b  4=  0. 

Equivalent  Inequations  of  Different  Forms.  —  The  laws  of  the  equiva- 
lence of  inequations  follow  immediately  from  their  analogues  for  equations. 

8-3     a  =}=  b  is  equivalent  to  a  -b  +  -a  b  =(=  0. 

[7-1] 
8-31     a  4=  1  is  equivalent  to  -a  4=  0. 

[7-12] 

8-32  If  3>(xi,  x<z,  ...  xn)  and  V(xi,  x2,  ...  xn)  be  any  two  functions  of 
the  same  variables,  then 


i,  X2,    ...    Xn~)    4=    ty(Xi,  X2,    ...    Xn) 

is  equivalent  to  F(x\,  x2,  ...  xn)  =j=  0,  where  F  is  a  function  of  these  same 
variables  and  such  that,  if  Ai,  A%,  A3,  etc.,  be  the  coefficients  in  $  and 
BI,  J52,  Bs,  etc.,  be  the  coefficients  of  the  corresponding  terms  in  M>,  then 
the  coefficients  of  the  corresponding  terms  in  F  will  be  A\  -Bi  +  -Ai  BI, 
Az  -B2  +  -A2  B2,  As  -B3  +  -As  B3>  etc. 

[7-13] 
Poretsky's  Law  of  Forms  for  inequations  will  be  : 

8-33     a  =J=  0  is  equivalent  to  t  4=  a  ~t+  -a  t. 

[7-151 
Or  in  more  general  form  : 

8  •  34     a  41  b  is  equivalent  to  t  +  (ab  +  -a  -6)  t  +  (a  -b  +  -a  b)  -t. 

[7-16] 

Elimination.  —  The  laws  governing  the  elimination  of  elements  from  an 
inequation  are  not  related  to  the  corresponding  laws  governing  equations 
by  the  reductio  ad  absurdum.  But  these  laws  follow  from  the  same  theorems 
concerning  the  limits  of  functions. 

8-4     If  Ax  +  B-x  4=  0,  then  ^4  +  5  +  0. 

[6  •  3]  A  x  +  B  -x  c  A  +  B.     Hence  [8  •  251  Q.E.D. 

8-41     If  the  coefficients  in  any  function  of  n  variables,  F(XI,  x2,  ...  #„), 


170  A  Survey  of  Symbolic  Logic 

be  Ci,  C'2,  Cz,  etc.,  and  if  F(xlt  .r2,  .  .  .  .rn)  =|=  0,  then 

EC*O 

[6-32]  F(xi,  .TO,  ...  xn)cC.     Hence  [8-25]  Q.E.D. 

Thus,  to  eliminate  any  number  of  elements  from  an  inequation  with 
one  member  0,  reduce  the  other  member  to  the  form  of  a  normal  function 
of  the  elements  to  be  eliminated.  The  elimination  is  then  secured  by 
putting  =(=  0  the  sum  of  the  coefficients.  The  form  of  elimination  resultants 
for  inequations  of  other  types  follows  immediately  from  the  above.  It  is 
obvious  that  they  will  be  analogous  to  the  elimination  resultants  of  equa- 
tions as  follows:  To  get  the  elimination  resultant  of  any  inequation,  take 
the  elimination  resultant  of  the  corresponding  equation  and  replace  =  by  4=  > 
and  x  by  +  . 

A  universal  proposition  in  logic  is  represented  by  an  equation:  "All 
a  is  b  "  by  a  -b  =  0,  "  No  a  is  b  "  by  a  b  =  0.  Since  a  particular  proposition 
is  always  the  contradictory  of  some  universal,  any  particular  proposition 
may  be  represented  by  an  inequation:  "Some  a  is  b"  by  a  b  =f=  0,  "Some 
a  is  not  b"  by  a  -b  =(=  0.  The  elimination  of  the  "middle"  term  from  an 
equation  which  represents  the  combination  of  two  universal  premises 
gives  the  equation  which  represents  the  universal  conclusion.  But  elimina- 
tion of  terms  from  inequations  does  not  represent  an  analogous  logical 
process.  Two  particulars  give  no  conclusion:  a  particular  conclusion 
requires  one  universal  premise.  The  drawing  of  a  particular  conclusion  is 
represented  by  a  process  which  combines  an  equation  with  an  inequation, 
by  8-17,  and  then  simplifies  the  result,  by  8-22.  For  example, 

All  a  is  b,        a  -b  =  0.         .'.  a  -b  c  =  0. 
Some  a  is  c,  a  c  +  0.         .'.  a  b  c  +  a  -b  c  ={=  0. 

.*.  abc  *  0.  [8-17] 

Some  b  is  c.  .'.  be  ^  0.  [8-22] 

"  Solution  "  of  an  Inequation. — An  inequation  may  be  said  to  have  a 
solution  in  the  sense  that  for  any  inequation  involving  x  an  equivalent 
inequation  one  member  of  which  is  x  can  always  be  found. 

8-5     A  x  +  B  -x  4=  0  is  equivalent  to  x  4=  -A  x  +  B  -x. 

[7-23] 

8-51     A  x  +  B  -x  =f=  0  is  equivalent  to  "Either  B-x^OorAx^O  ",— 
i.  e.,  to  "Either  B  ex  is  false  or  x  c-A  is  false  ". 

[7-2! 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  171 

Neither  of  these  "solutions"  determines  x  even  within  limits.  "Bex 
is  false  "  does  not  mean  " B  is  excluded  from  x";  it  means  only  " B  is  not 
wholly  within  x  ".  "Either  Bex  is  false  or  xc-A  is  false"  does  not 
determine  either  an  upper  or  a  lower  limit  of  x;  and  limits  x  only  by  ex- 
cluding B  +  u  -A  from  the  range  of  its  possible  values.  Thus  "  solutions  " 
of  inequations  are  of  small  significance. 

Consequences  and  Sufficient  Conditions  of  an  Inequation. — By  Poret- 
sky's  method,  the  formula  for  any  consequence  of  a  given  inequation  follows 
from  the  Law  of  Sufficient  Conditions  for  equations.17  If  for  some  value 
of  u  and  some  value  of  v, 

t  =  v  (a  b  +  -a  -6)  t  +  (a  -b  +  -a  b  +  u)  -t 
then  a  =  b.     Consequently,  we  have  by  the  reductio  ad  absurdum: 

8-52     If    a  =(=  6,    then    t  ^  v  (ab  +  -a  -b)  t  +  (a  -b  +  -a  b  +  u)  -t,    where    u 
and  v  are  arbitrary. 

[7-7] 

The  formula  for  the  sufficient  conditions  of  an  inequation  similarly  fol- 
lows from  the  Law  of  Consequences  for  equations.  If  a  =  b,  then 

t  =  (a  b  +  -a  -b  +  u)  t  +  v  (a  -b  +  -a  b)  -t 

where  u  and  v  are  arbitrary.     Consequently,  by  the  reductio  ad  absurdum: 
8-53     If  for  some  value  of  u  and  some  value  of  v, 

t  =^  (a  b  +  -a  -b  +  u)  t  +  v  (a  -b  +  -a  6)  -t 
then  a  =(=  b. 

[7-6] 

System  of  an  Equation  and  an  Inequation. — If  we  have  an  equation  in 
one  unknown,  x,  and  an  inequation  which  involves  x,  these  may  be  combined 
in  either  of  two  ways:  (1)  each  may  be  reduced  to  the  form  in  which  one 
member  is  0  and  expanded  with  reference  to  all  the  elements  involved  in 
either.  Then  all  the  terms  which  are  common  to  the  two  may,  by  8-17, 
be  dropped  from  the  inequation;  (2)  the  equation  may  be  solved  for  x, 
and  this  value  substituted  for  x  in  the  inequation. 

8-6     If  A  x  +  B  -x  =  0  and  C x  +  D  -x  4=  0,  then  -A  C  x  +  -B  D  -x  4=  0. 
[5-8]  If  Cx  +  D-x  4=  0,  then 

A  C  x  +  -A  C  x  +  B  D  -x  +  -B  D  -x  4=  0 

17  See  Poretsky,  Theorie  des  non-egalites  logiques,  Chaps.  71,  76. 


172  A  Survey  of  Symbolic  Logic 

[5-72]  If  Ax  +  B-x  =  0,  then  A  x  =  0  and  B  -x  =  0,  and  hence 

A  C  x  =  0  and  B  D  -x  =  0. 

Hence  [8-17]  -A  C  x  +  -B  D  -x  4=  0. 

The  result  here  is  not  equivalent  to  the  data,  since — for  one  reason— 
the  equation  ACx  +  BD-x  =  0  is  not  equivalent  to  A  x  +  B  -x  =  0. 
Nevertheless  this  mode  of  combination  is  the  one  most  frequently  useful. 

8-61  The  condition  that  the  equation  Ax  +  B-x  =  0  and  the  inequation 
C  x  +  D  -x  4=  0  may  be  regarded  as  simultaneous  is,  A  B  =  0  and  -A  C 
+  -B  D  4=  0,  and  the  determination  of  x  which  they  give  is 

x  4=  (-A  -C  +  A-D)x  +  (BC  +  -B  D)  -x 

[7-23]   A  x  +  B  -x  =  0  is  equivalent  to  x  =  -A  x  +  B  -x.     Substi- 
tuting this  value  of  x  in  the  inequation, 

C  (-A  x  +  B  -x)  +  D  (A  x  +  -B  -x}  4=  0 

or  (-AC  +  AD)x+(BC  +  -BD)-x  +  0. 
[8-4]  A  condition  of  this  inequation  is 

(-AC  +  AD)  +  (BC  +  -BD)  4=  0, 

or  (-A+B)  C  +  (A  +  -B)  D  =j=  0. 

But  the  equation  A  x  +  B  -x  =  0  requires  that  A  B  =  0,  and  hence 

that  -A  +  B  =  -A  and  -B  +  A  =  -B. 

Hence  if  the  equation  be  possible  and  A  B  =  0,  the  condition  of  the 

inequation  reduces  to  -A  C  +  -B  D  4=  0. 

[8-4]  If  the  original  inequation  be  possible,  then  C  +  D  4=  0.     But 

this  condition  is  already  present  in  -A  C  +  -B  D  4=  0,  since  -A  C  cC 

and   hence   [8-25]   if  -A  C  4=  0,   then   (7  +  0,   and   -BDcD   and 

hence  if  -B  D  4=  0,  then  D  4=  0,  while  [8-26]  C  +  D  4=  0  is  equivalent 

to  "  Either  C  4=  0  or  D  4=  0  ",  and  -A  C  +  -B  D  4=  0  is  equivalent  to 

"  Either  -A  C  4=  0  or  -B  D  4=  0  ". 

Hence  the  entire  condition  of  the  system  is  expressed  by 

AB  =  0        and        -AC  +  -BD*Q 
And  [8-5]  the  solution  of  the  inequation, 

(-AC  +  AD)x  +  (BC  +  -BD)-x  4=  0,        is 

x  4=  (-A-C  +  A-D)x  +  (BC  +  -BD)-x 

This  method  gives  the  most  complete  determination  of  x,  in  the  form  of 
an  inequation,  afforded  by  the  data. 


The  Classic,  or  Boole-Schroder,  Algebra  of  Logic  173 

VI.    NOTE  ON  THE  INVERSE  OPERATIONS,  "SUBTRACTION"  AND 

"DIVISION" 

It  is  possible  to  define  "subtraction"  {  — }  and  "division"  {  :  }  in  the 
algebra.  Let  a  —  b  be  x  such  that  b  +  x  =  a.  And  let  a  :  b  be  y  such 
that  b  y  =  a.  However,  these  inverse  operations  are  more  trouble  than 
they  are  worth,  and  should  not  be  admitted  to  the  system. 

In  the  first  place,  it  is  not  possible  to  give  these  relations  a  general 
meaning.  We  cannot  have  in  the  algebra:  (1)  If  a  and  b  are  elements 
in  K,  then  a  :  b  is  an  element  in  K;  nor  (2)  If  a  and  b  are  elements  in  K, 
then  a  —  b  is  an  element  in  K.  If  a  :  b  is  an  element,  y,  then  for  some  y 
it  must  be  true  that  b  y  =  a.  But  if  b  y  =  a,  then,  by  2 •  2,  acby  and, 
by  5-2,  a  c  b.  Thus  if  a  and  b  be  so  chosen  that  a  c  b  is  false,  then  a  :  6 
cannot  be  any  element  in  K.  To  give  a  :  b  a  general  meaning,  it  would 
be  required  that  every  element  be  contained  in  every  element — that  is, 
that  all  elements  in  K  be  identical.  Similarly,  if  a  —  b  be  an  element, 
x,  in  K,  then  for  some  x,  it  must  be  true  that  b  +  x  =  a.  But  if  b  +  x  =  a, 
then,  by  2-2,  b  +  xca  and,  by  5-21,  be  a.  Thus  if  a  and  b  be  so  chosen 
that  &  c  a  is  false,  then  a  —  b  cannot  be  any  element  in  K. 

Again,  a  —  b  and  a  :  b  are  ambiguous.  It  might  be  expected  that, 
since  a  +  -a  =  1,  the  value  of  1  —  a  would  be  unambiguously  -a.  But 
1  —  a  =  x  is  satisfied  by  any  x  such  that  -a  c  x.  For  1  —  a  =  x  is  equiva- 
lent to  x  +  a  =  1,  which  is  equivalent  to 

-(x  +  a)  =  -1  =  0  =  -a  -x 

And  -a  -x  =  0  is  equivalent  to  -a  c  x.  Similarly,  it  might  be  expected 
that,  since  a  -a  =  0,  the  value  of  0  :  a  would  be  unambiguously  -a.  But 
0  :  a  =  y,  or  a  y  =  0,  is  satisfied  by  any  y  such  that  y  c  -a.  a  y  =  0  and 
y  c  -a  are  equivalent. 

Finally,  these  relations  can  always  be  otherwise  expressed.  The  value 
of  a  :  b  is  the  value  of  y  in  the  equation,  b  y  =  a.  b  y  =  a  is  equivalent  to 

-a  b  y  +  a  -b  +  a  -y  =  0 

The  equation  of  condition  here  is  a  -b  —  0.  And  the  solution,  on  this 
condition,  is 

y  =  a  +  u  (a  +  -&)  =  ab  +  u-a-b,  where  u  is  undetermined. 

The  value  of  a  —  b  is  the  value  of  x  in  the  equation,  6  +  x  =  a.  b  +  x  =  a 
is  equivalent  to 

-a  b  +  -a  x  +  a  -b  -x  =  0 


174  A  Survey  of  Symbolic  Logic 

The  equation  of  condition  here  is,  -a  b  =  0.     And  the  solution,  on  this 
condition,  is 

x  =  a-b  +  v  a  =  a-b  +  v  ab,  where  v  is  undetermined. 

In  each  case,  the  equation  of  condition  gives  the  limitation  of  the  meaning 
of  the  expression,  and  the  solution  expresses  the  range  of  its  possible  values. 


CHAPTER    III 

APPLICATIONS  OF  THE  BOOLE-SCHRODER  ALGEBRA 

There  are  four  applications  of  the  classic  algebra  of  logic  which  are 
commonly  considered:  (1)  to  spatial  entities,  (2)  to  the  logical  relations 
of  classes,  (3)  to  the  logical  relations  of  propositions,  (4)  to  the  logic  of 
relations. 

The  application  to  spatial  entities  may  be  made  to  continuous  and 
discontinuous  segments  of  a  line,  or  to  continuous  and  discontinuous  regions 
in  a  plane,  or  to  continuous  and  discontinuous  regions  in  space  of  any 
dimensions.  Segments  of  a  line  and  regions  in  a  plane  have  both  been 
used  as  diagrams  for  the  relations  of  classes  and  of  propositions,  but  the 
application  to  regions  in  a  plane  gives  the  more  workable  diagrams,  for 
obvious  reasons.  And  since  it  is  only  for  diagrammatic  purposes  that 
the  application  of  the  algebra  to  spatial  entities  has  any  importance,  we 
shall  confine  our  attention  to  regions  in  a  plane. 

I.    DIAGRAMS  FOR  THE  LOGICAL  RELATIONS  OF  CLASSES 

For  diagrammatic  purposes,  the  elements  of  the  algebra,  a,  b,  c,  etc., 
will  denote  continuous  or  discontinuous  regions  in  a  given  plane,  or  in  a 
circumscribed  portion  of  a  plane.  1  represents  the  plane  (or  circumscribed 
portion)  itself.  0  is  the  null-region  which  is  supposed  to  be  contained  in 
every  region.  For  any  given  region,  a,  -a  denotes  the  plane  exclusive  of 
a, — i.  e.,  not-a.  The  "product",  a  x6  or  a  b,  is  that  region  which  is  com- 
mon to  a  and  b.  If  a  and  b  do  not  "overlap  ",  then  a  fy  is  the  null-region,  0. 
The  "sum",  a  +  b,  denotes  the  region  which  is  either  a  or  b  (or  both).  In 
determining  a  +  b,  the  common  region,  a  b,  is  not,  of  course,  counted  twice 
over. 

a  +  b  =  a-b  +  ab  +  -ab. 

This  is  a  difference  between  +  in  the  Boole-Schroder  Algebra  and  the  + 
of  arithmetic.  The  equation,  a  =  b,  signifies  that  a  and  6  denote  the  same 
region,  a  c  b  signifies  that  a  lies  wholly  within  b,  that  a  is  included  or 
contained  in  b.  It  should  be  noted  that  whenever  a  =  b,  a  c  b  and  b  c  a. 
Also,  a  c  a  holds  always.  Thus  the  relation  c  is  analogous  not  to  <  in 

arithmetic  but  to  — . 

175 


176  A  Survey  of  Symbolic  Logic 

While  the  laws  of  this  algebra  hold  for  regions,  thus  denoted,  however 
those  regions  may  be  distributed  in  the  plane,  not  every  supposition  about 
their  distribution  is  equally  convenient  as  a  diagram  for  the  relations  of 
classes.  All  will  be  familiar  with  Euler's  diagrams,  invented  a  century 
earlier  than  Boole's  algebra.  "All  a  is  b"  is  represented  by  a  circle  a 
wholly  within  a  circle  b;  "  No  a  is  b  "  by  two  circles,  a  and  b,  which  nowhere 
intersect;  "Some  a  is  b"  and  "Some  a  is  not  b"  by  intersecting  circles, 
sometimes  with  an  asterisk  to  indicate  that  division  of  the  diagram  \vhich 
represents  the  proposition.  The  defects  of  this  style  of  diagram  are  obvious: 


All  a  is  6  No  a  is  b  Some  a  is  6  Some  a  is  not  b 

FIG.  1 

the  representation  goes  beyond  the  relation  of  classes  indicated  by  the  propo- 
sition. In  the  case  of  "All  a  is  b",  the  circle  a  falls  within  b  in  such  wise 
as  to  suggest  that  we  may  infer  "Some  b  is  not  a",  but  this  inference  is 
not  valid.  The  representation  of  "No  a  is  b"  similarly  suggests  "Some 
things  are  neither  a  nor  b",  'which  also  is  unwarranted.  With  these  dia- 
grams, there  is  no  way  of  indicating  whether  a  given  region  is  null.  But 
the  general  assumption  that  no  region  of  the  diagram  is  null  leads  to  the 
misinterpretations  mentioned,  and  to  others  which  are  similar.  Yet 
Euler's  diagrams  were  in  general  use  until  the  invention  of  Venn,  and  are 
still  doing  service  in  some  quarters. 

The  Venn  diagrams  were  invented  specifically  to  represent  the  relations 
of  logical  classes  as  treated  in  the  Boole-Schroder  Algebra.1  The  principle 
of  these  diagrams  is  that  classes  be  represented  by  regions  in  such  relation 
to  one  another  that  all  the  possible  logical  relations  of  these  classes  can  be 
indicated  in  the  same  diagram.  That  is,  the  diagram  initially  leaves  room 
for  any  possible  relation  of  the  classes,  and  the  actual  or  given  relation  can 
then  be  specified  by  indicating  that  some  particular  region  is  null  or  is  not- 
null.  Initially  the  diagram  represents  simply  the  "universe  of  discourse", 
or  1.  For  one  element,  a,  1  —  a  +  -a.2  For  two  elements,  a  and  b, 

1  =  (a  +  -a)  (6  +  -6)  =  a  b  +  a  -b  +  -a  b  +  -a  -b 

1  See  Venn,  Symbolic  Logic,  Chap.  v.     The  first  edition  of  this  book  appeared  before 
Schroder's  Algebra  der  Logik,  but  Venn  adopts  the  most  important  alteration  of  Boole's 
original  algebra — the  non-exclusive  interpretation  of  a  +  b. 

2  See  above,  Chap,  u,  propositions  4-8  and  5-92. 


Applications  of  the  Boole-Schroder  Algebra 


177 


For  three  elements,  a,  b,  and  c, 
1  =  (a  +  -a)  (b  +  -6)  (c  +  -c)  =  a 


a-bc+-abc  +  a-b-c 

+  -a  b  -c  +  -a  -b  c  +  -a  -b  -c 


Thus  the  "universe  of  discourse"  for  any  number  of  elements,  n,  must 
correspond  to  a  diagram  of  2"  divisions,  each  representing  a  term  in  the 
expansion  of  1.  If  the  area  within  the  square  in  the  diagram  represent 


-a 


-a-b 


FIG.  2 

the  universe,  and  the  area  within  the  circle  represent  the  element  a,  then 
the  remainder  of  the  square  will  represent  its  negative,  -a.  If  another 
element,  b,  is  to  be  introduced  into  the  same  universe,  then  b  may  be  repre- 
sented by  another  circle  whose  periphery  cuts  the  first.  The  divisions, 
(1)  into  a  and  -a,  (2)  into  b  and  -b,  will  thus  be  cross-divisions  in  the  uni- 
verse. If  a  and  b  be  classes,  this  arrangement  represents  all  the  possible 
subclasses  in  the  universe; — a  b,  those  things  which  are  both  a  and  b; 
a  -b,  those  things  which  are  a  but  not  b ;  -ab,  those  things  which  are  b 
but  not  a;  -a  -b,  those  things  which  are  neither  a  nor  6.  The  area  which 
represents  the  product,  a  b,  will  readily  be  located.  We  have  enclosed 
by  a  broken  line,  in  figure  2,  the  area  which  represents  a  +  b. 

The  negative  of  any  entity  is  always  the  plane  exclusive  of  that  entity. 
For  example,  -(a  b  +  -a  -6),  in  the  above,  will  be  the  sum  of  the  other 
two  divisions  of  the  diagram,  a-b  +  -a  b. 

If  it  be  desired  t®  introduce  a  third  element,  c,  into  the  universe,  it  is 
necessary  to  cut  each  one  of  the  previous  subdivisions  into  two — one 
part  which  shall  be  in  c  and  one  part  which  shall  be  outside  c.  This  can  be 
be  accomplished  by  introducing  a  third  circle,  thus 

It  is  not  really  necessary  to  draw  the  square,  1,  since  the  area  given  to  the 
figure,  or  the  whole  page,  may  as  well  be  taken  to  represent  the  universe. 
But  when  the  square  is  omitted,  it  must  be  remembered  that  the  unenclosed 
13 


178 


A  Survey  of  Symbolic  Logic 


area  outside  all  the  lines  of  the  figure  is  a  subdivision  of  the  universe — 
the  entity  -a,  or  -a  -6,  or  -a  -b  -c,  etc.,  according  to  the  number  of  elements 
involved. 


-a-b-c 


FIG.  3 


If  a  fourth  element,  d,  be  introduced,  it  is  no  longer  possible  to  repre- 
sent each  element  by  a  circle,  since  a  fourth  circle  could  not  be  introduced 
in  figure  3  so  as  to  cut  each  previous  subdivision  into  two  parts — one  part 
in  d  and  one  part  outside  d.  But  this  can  be  done  with  ellipses.3  Each 


FIG.  4 

3  We  have  deformed  the  ellipses  slightly  and  have  indicated  the  two  points  of  junction. 
This  helps  somewhat  in  drawing  the  diagram,  which  is  most  easily  done  as  follows:  First, 
draw  the  upright  ellipse,  a.  Mark  a  point  at  the  base  of  it  and  one  on  the  left.  Next, 


Applications  of  the  Boole-Schroder  Algebra 


179 


one  of  the  subdivisions  in  figure  4  can  be  "named"  by  noting  whether  it 
is  in  or  outside  of  each  of  the  ellipses  in  turn.  Thus  the  area  indicated  by 
6  is  a  b  c  -d,  and  the  area  indicated  by  12  is  -a  -b  c  d.  With  a  diagram  of 
four  elements,  it  requires  care,  at  first,  to  specify  such  regions  as  a  +  c, 
a  c  +  b  d,  b  +  -d.  These  can  always  be  determined  with  certainty  by 
developing  each  term  of  the  expression  with  reference  to  the  missing  ele- 
ments.4 Thus 


ac  +  b  d  =  ac  (b  +  -b)  (d  +  -d)  +  b  d  (a  +  -a)  (c  +  -c) 
=  abcd+abc-d  +  a-bcd+a-bc-d  +  ab 


-ab  -c  d 


The  terms  of  this  sum,  in  the  order  given,  are  represented  in  figure  4  by  the 
divisions  numbered  10,  6,  9,  5,  14,  11,  15.  Hence  ac  +  bd  is  the  region 
which  combines  these.  With  a  little  practice,  one  may  identify  such 
regions  without  this  tedious  procedure.  Such  an  area  as  b  +  -d  is  more 
easily  identified  by  inspection:  it  comprises  2,  3,  6,  7,  10,  11,  14,  15,  and 
1,  4,  5,  8. 

Into  this  diagram  for  four  elements,  it  is  possible  to  introduce  a  fifth, 
e,  if  we  let  e  be  the  region  between  the  broken  lines  in  figure  5.  The  principle 
of  the  "  square  diagram"  (figure  6)  is  the  same  as  Venn's:  it  represents  all 


FIG.  5 

draw  the  horizontal  ellipse,  d,  from  one  of  these  points  to  the  other,  so  that  the  line  con- 
necting the  two  points  is  common  to  a  and  d.  Then,  draw  ellipse  b  from  and  returning  to 
the  base  point,  and  ellipse  c  from  and  returning  to  the  point  on  the  left.  If  not  done  in 
this  way,  the  first  attempts  are  likely  to  give  twelve  or  fourteen  subdivisions  instead  of 
the  required  sixteen. 
4  See  Chap,  u,  5-91. 


180 


A  Survey  of  Symbolic  Logic 


the  subclasses  in  a  universe  of  the  specified  number  of  elements.     No 
diagram  is  really  convenient  for  more  than  four  elements,  but  such  are 


5 

-b 

a~b 

-ab 

a—I) 

—a  —b 

FIG.  6 


frequently  needed.     The  most  convenient  are  those  made  by  modifying 
slightly  the  square  diagram  of  four  terms,  at  the  right  in  figure  6.5    Figure  7 


-a 


-e 


-h 
FIG.  7 


-e 


gives,  by  this  method,  the  diagrams  for  five  and  for  six  elements.  We  give 
also  the  diagram  for  seven  (figure  8)  since  this  is  frequently  useful  and 
not  easy  to  make  in  any  other  way. 

The  manner  in  which  any  function  in  the  algebra  may  be  specified  in  a 
diagram  of  the  proper  number  of  divisions,  has  already  been  explained. 
We  must  now  consider  how  any  asserted  relation  of  elements — any  inclu- 

6  See  Lewis  Carroll,  Symbolic  Logic,  for  the  particular  form  of  the  square  diagram  which 
we  adopt.  Mr.  Dodgson  is  able,  by  this  method,  to  give  diagrams  for  as  many  as  10  terms, 
1024  subdivisions  (p.  176). 


Applications  of  the  Boole-Schroder  Algebra 


181 


sion,  a  c  b,  or  any  equation,  a  =  b,  or  inequation,  a  +  b — may  be  repre- 
sented. Any  such  relation,  or  any  set  of  such  relations,  can  be  completely 
specified  in  these  diagrams  by  taking  advantage  of  the  fact  that  they 


-e 


FIG.  8 

can  always  be  reduced  either  to  the  form  of  an  expression  =  0  or  to  the 
form  of  an  expression  =|=  0.  Any  inclusion,  a  c  b,  is  equivalent  to  an  equa- 
tion, a  -b  =  O.6  And  every  equation  of  the  form  a  =  b  is  equivalent  to 
one  of  the  form  a-b  +  -ab  =  O.7  Thus  any  inclusion  or  equation  can  be 
represented  by  some  expression  =  0.  Similarly,  any  inequation  of  the 
form  a  =f=  b  is  equivalent  to  one  of  the  form  a-b  +  -ab  =j=  O.8  Thus  any 
asserted  relation  whatever  can  be  specified  by  indicating  that  some  region 
(continuous  or  discontinuous)  either  is  null,  {  =  0},  or  is  not-null,  {4=  0}. 
We  can  illustrate  this,  and  at  the  same  time  indicate  the  manner  in 
which  such  diagrams  are  useful,  by  applying  the  method  to  a  few  syllogisms* 


Given :  All  a  is  b, 
and  All  b  is  c, 

8  See  Chap.  11,  4-9. 

7  See  Chap,  n,  6-4. 

8  See  Chap,  n,  7-1. 


a  cb, 
b  cc, 


a  -b  =  0. 
b  -c  =  0. 


182 


A  Survey  of  Symbolic  Logic 


We  have  here  indicated  (figure  9)  that  a  -b — the  a  which  is  not  b — is  null 
by  striking  it  out  (with  horizontal  lines).  Similarly,  we  have  indicated 
that  all  b  is  c  by  striking  out  b  -c  (with  vertical  lines).  Together,  the  two 
operations  have  eliminated  the  whole  of  a  -c,  thus  indicating  that  a  -c  =  0, 
or  "All  aisc". 


FIG.  9 


FIG.  10 


For  purposes  of  comparison,  we  may  derive  this  same  conclusion  by 
algebraic  processes.9 

Since  a  -b  =  0  =  a  -b  (c  +  -c)  =  a  -b  c  +  a  -b  -c, 

and  b  -c  =  0  =  b  -c  (a  +  -a)  =  a  b  -c  +  -a  b  -c, 
therefore,  a  -b  c  +  a  -b  -c  +  a  b  -c  +  -a  b  -c  =  0, 

and  [5-72]  a  b  -c  +  a  -b  -c  =  0  =  a  -c  (b  +  -b)  =  a  -c. 

The  equation  in  the  third  line,  which  combines  the  two  premises,  states 
exactly  the  same  facts  which  are  represented  in  the  diagram.  The  last 
equation  gives  the  conclusion,  which  results  from  eliminating  the  middle 
term,  b.  Since  a  diagram  will  not  perform  an  elimination,  we  must  there 
"look  for"  the  conclusion. 

One  more  illustration  of  this  kind : 

Given:  All  a  is  b,        a-b  =  0. 
and  No  6  is  c,  b  c  =  0. 

The  first  premise  is  indicated  (figure  10)  by  striking  out  the  area  a  -b  (with 
horizontal  lines),  the  second  by  striking  out  be  (with  vertical  lines).  To- 
gether, these  operations  have  struck  out  the  whole  of  a  c,  giving  the  con- 
clusion a  c  =  0,  or  "No  a  is  c". 

9  Throughout  this  chapter,  references  in  square  brackets  give  the  number  of  the  the- 
orem in  Chap,  n  by  which  any  unobvious  step  in  proof  is  taken. 


Applications  of  the  Boole-Schroder  Algebra 


183 


In  a  given  diagram  where  all  the  possible  classes  or  regions  in  the  uni- 
verse are  initially  represented,  as  they  are  by  this  method  of  diagramming, 
we  cannot  presume  that  a  given  subdivision  is  null  or  is  not-null.  The 
actual  state  of  affairs  may  require  that  some  regions  be  null,  or  that  some 
be  not-null,  or  that  some  be  null  and  others  not.  Consequently,  even 
when  we  have  struck  out  the  regions  which  are  null,  we  cannot  presume 
that  all  the  regions  not  struck  out  are  not-null.  This  would  be  going  beyond 
the  premises.  All  we  can  say,  when  we  have  struck  out  the  null-regions, 
is  that,  so  far  as  the  premises  represented  are  concerned,  any  region  not 
struck  out  may  be  not-null.  If,  then,  we  wish  to  represent  the  fact  that  a 
given  region  is  definitely  not-null — that  a  given  class  has  members,  that 
there  is  some  expression  =}=  0 — we  must  indicate  this  by  some  distinctive 
mark  in  the  diagram.  For  this  purpose,  it  is  convenient  to  use  asterisks. 
That  a  b  =}=  0,  may  be  indicated  by  an  asterisk  in  the  region  a  b.  But  here 
a  further  difficulty  arises.  If  the  diagram  involve  more  than  two  elements, 
say,  a,  b,  and  c,  the  region  a  b  will  be  divided  into  two  parts,  a  b  c  and 
a  b  -c.  Now  the  inequation,  a  b  =f=  0,  does  not  tell  us  that  a  b  c  +  0,  and 
it  does  not  tell  us  that  a  b  -c  =t=  0.  It  tells  us  only  that  ab  c  +  ab  -c  4=  0. 
If,  then,  we  wish  to  indicate  a  b  =}=  0  by  an  asterisk  in  the  region  a  b,  we 
shall  not  be  warranted  in  putting  it  either  inside  the  circle  c  or  outside  c. 
It  belongs  in  one  or  the  other  or  both — that  is  all  we  know.  Hence  it  is 
convenient  to  indicate  a  b  4=  0  by  placing  an  asterisk  in  each  of  the  divisions 
of  a  6  and  connecting  them  by  a  broken  line,  to  signify  that  at  least  one  of 
these  regions  is  not-null  (figure  11). 


FIG.  11 


We  shall  show  later  that  a  particular  proposition  is  best  interpreted  by 
an  inequation;    "Some  a  is  b",  the  class  ab  has  members,  by  a  b  4=  0. 


184  A  Survey  of  Symbolic  Logic 

Suppose,  then,  we  have: 

Given:     All  a  is  b,  a-b  =  0. 

and  Some  a  is  c,  a  c  =|=  0. 

The  conclusion,  "Some  b  is  c",  is  indicated  (figure  12)  by  the  fact  that 
one  of  the  two  connected  asterisks  must  remain — the  whole  region  a  b  c 
+  a-bc  cannot  be  null.  But  one  of  them,  in  a  -b  c,  is  struck  out  in  indi- 
cating the  other  premise,  a  -b  =  0.  Thus  a  6  c  4=  0,  and  hence  a  c  4=  0. 


FIG.  12 

The  entire  state  of  affairs  in  a  universe  of  discourse  may  be  represented 
by  striking  out  certain  regions,  indicating  by  asterisks  that  certain  regions 
are  not-null,  and  remembering  that  any  region  which  is  neither  struck 
out  nor  occupied  by  an  asterisk  is  in  doubt.  Also,  the  separate  subdivisions 
of  a  region  occupied  by  connected  asterisks  are  in  doubt  unless  all  but  one 
of  these  connected  asterisks  occupy  regions  which  are  struck  out.  And 
any  regions  which  are  left  in  doubt  by  a  given  set  of  premises  might,  of 
course,  be  made  specifically  null  or  not-null  by  an  additional  premise. 

In  complicated  problems,  the  use  of  the  diagram  is  often  simpler  and 
more  illuminating  than  the  use  of  transformations,  eliminations,  and  solu- 
tions in  the  algebra.  All  the  information  to  be  derived  from  such  opera- 
tions, the  diagram  gives  (for  one  who  can  "see"  it)  at  a  glance.  Further 
illustrations  will  be  unnecessary  here,  since  we  shall  give  diagrams  in  con- 
nection with  the  problems  of  the  next  section. 

II.    THE  APPLICATION  TO  CLASSES 

The  interpretation  of  the  algebra  for  logical  classes  has  already  been 
explained.10    a,  b,  c,  etc.,  are  to  denote  classes  taken  in  extension;  that  is 
10  Chap,  n,  pp.  121-22. 


Applications  of  the  Boole-Schroder  Algebra  185 

to  say,  c  signifies,  not  a  class-concept,  but  the  aggregate  of  all  the  objects 
denoted  by  some  class-concept.  Thus  if  a  =  b,  the  concept  of  the  class  a 
may  not  be  a  synonym  for  the  concept  of  the  class  b,  but  the  classes  a  and  b 
must  consist  of  the  same  members,  have  the  same  extension,  a  c  b  sig- 
nifies that  every  member  of  the  class  a  is  also  a  member  of  the  class  6. 
The  "product",  a  6,  denotes  the  class  of  those  things  which  are  both  mem- 
bers of  a  and  members  of  6.  The  "sum  ",  a  +  b,  denotes  the  class  of  those 
things  which  are  either  members  of  a  or  members  of  b  (or  members  of  both). 
0  denotes  the  null-class,  or  class  without  members.  Various  concepts  may 
denote  an  empty  class — "immortal  men",  "feathered  invertebrates", 
"Julius  Caesar's  twin,"  etc. — but  all  such  terms  have  the  same  extension; 
they  denote  nothing  existent.  Thus,  since  classes  are  taken  in  extension, 
there  is  but  one  null-class,  0.  Since  it  is  a  law  of  the  algebra  that,  for 
every  x,  0  ex,  we  must  accept,  in  this  connection,  the  convention  that 
the  null-class  is  contained  in  every  class.  All  the  immortal  men  are  mem- 
bers of  any  class,  since  there  are  no  such.  1  represents  the  class  "every- 
thing ",  the  "universe  of  discourse  ",  or  simply  the  "universe  ".  This  term 
is  pretty  well  understood.  But  it  may  be  defined  as  follows:  if  an  be  any 
member  of  the  class  a,  and  X  represent  the  class-concept  of  the  class  x, 
then  the  "universe  of  discourse"  is  the  class  of  all  the  classes,  x,  such  that 
"an  is  an  X"  is  either  true  or  false.  If  "The  fixed  stars  are  blind"  is 
neither  true  nor  false,  then  "fixed  stars"  and  the  class  "blind"  do  not 
belong  to  the  same  universe  of  discourse. 

The  negative  of  a,  -a,  is  a  class  such  that  a  and  -a  have  no  members  in 
common,  and  a  and  -a  between  them  comprise  everything  in  the  universe 
of  discourse:  a  -a  =  0,  "Nothing  is  both  a  and  not-a",  and  a  +  -a  =  1, 
"Everything  is  either  a  or  -a". 

Since  inclusions,  a  c  b,  equations,  a  =  b,  and  inequations,  a  =(=  b,  repre- 
sent relations  which  are  asserted  to  hold  between  classes,  they  are  capable 
of  being  interpreted  as  logical  propositions.  And  the  operations  of  the 
algebra — transformations,  eliminations,  and  solutions — are  capable  of 
interpretation  as  processes  of  reasoning.  It  would  hardly  be  correct  to 
say  that  the  operations  of  the  algebra  represent  the  processes  of  reasoning 
from  given  premises  to  conclusions:  they  do  indeed  represent  processes 
of  reasoning,  but  they  seldom  attain  the  result  by  just  those  operations 
which  are  supposed  to  characterize  the  customary  processes  of  thinking. 
In  fact,  it  is  the  greater  generality  of  the  symbolic  operations  which  makes 
their  application  to  reasoning  valuable. 


186  A  Survey  of  Symbolic  Logic 

The  representation  of  propositions  by  inclusions,  equations,  and  in- 
equations, and  the  interpretation  of  inclusions,  equations,  and  inequations 
in  the  algebra  as  propositions,  offers  certain  difficulties,  due  to  the  fact 
that  the  algebra  represents  relations  of  extension  only,  while  ordinary  logical 
propositions  quite  frequently  concern  relations  of  intension.  In  discussing 
the  representation  of  the  four  typical  propositions,  we  shall  be  obliged  to 
consider  some  of  these  problems  of  interpretation. 

The  universal  affirmative,  "All  a  is  b",  has  been  variously  represented  as, 

(1)  a  =  a  b, 

(2)  acb, 

(3)  a  =  v  b,  where  v  is  undetermined, 

(4)  a  -b  =  0. 

All  of  these  are  equivalent.11  The  only  possible  doubt  concerns  (3)  a  =  v  b, 
where  v  is  undetermined.  But  its  equivalence  to  the  others  may  be  demon- 
strated as  follows: 

[7  •  1]  a  =  v  b  is  equivalent  to  a  •  -(v  b)  +  -a  v  b  =  0. 

But  a--(v  6)  +  -a  v  b  =  a  (-v  +  -b)  +  -avb  =  a  -v  +  a  -b  +  -a  %  b. 

Hence  [5-72]  if  a  =  v  b,  then  a  -b  =  0. 

And  if  a  —  a  b,  then  for  some  value  of  v  (i.  e.,  v  =  a),  a  =  v  b. 

These  equivalents  of  "  All  a  is  b  "  would  most  naturally  be  read : 

(1)  The  a's  are  identical  with  those  things  which  are  a's  and  6's  both. 

(2)  a  is  contained  in  b:  every  member  of  a  is  also  a  member  of  b. 

(3)  The  class  a  is  identical  with  some  (undetermined)  portion  of  the 
class  6. 

(4)  The  class  of  those  things  which  are  members  of  a  but  not  members 
of  b  is  null. 

If  we  examine  any  one  of  these  symbolic  expressions  of  "All  a  is  b", 
we  shall  discover  that  not  only  may  it  hold  when  a  =  0,  but  it  always 
holds  when  a  =  0.  0  =  0-6,  0  cb,  and  0--6  =  0,  will  be  true  for  every 
element  b.  And  "0  =  06  for  some  value  of  v"  is  always  true — for  v  =  0. 
Since  a  =  0  means  that  a  has  no  members,  it  is  thus  clear  that  the  algebra 
requires  that  "All  a  is  b"  be  true  whenever  no  members  of  a  exist.  The 
actual  use  of  language  is  ambiguous  on  this  point.  We  should  hardly  say 
that  "All  sea  serpents  have  red  wings,  because  there  aren't  any  sea  ser- 
pents"; yet  we  understand  the  hero  of  the  novel  who  asserts  "Whoever 

11  See  Chap,  n,  4-9. 


Applications  of  the  Boole-Schroder  Algebra  187 

enters  here  must  pass  over  my  dead  body".  This  hero  does  not  mean  to 
assert  that  any  one  will  enter  the  defended  portal  over  his  body:  his  desire 
is  that  the  class  of  those  who  enter  shall  be  null.  The  difference  of  the 
two  cases  is  this:  the  concept  "sea  serpent"  does  not  necessarily  involve  the 
concept  "having  red  wings",  while  the  concept  of  "those  who  enter  the 
portal" — as  conceived  by  the  hero — does  involve  the  concept  of  passing 
over  his  body.  We  readily  accept  and  understand  the  inclusion  of  an 
empty  class  in  some  other  when  the  concept  of  the  one  involves  the  concept 
of  the  other — when  the  relation  is  one  of  intension.  But  in  this  sense,  an 
empty  class  is  not  contained  in  any  and  every  class,  but  in  some  only.  In 
order  to  understand  this  law  of  the  algebra,  "For  every  x,  0  ex",  we  must 
bear  in  mind  two  things:  (1)  that  the  algebra  treats  of  relations  in  extension 
only,  and  (2)  that  ordinary  language  frequently  concerns  relations  of 
intension,  and  is  usually  confined  to  relations  of  intension  where  a  null 
class  is  involved.  The  law  does  not  accord  with  the  ordinary  use  of  language. 
This  is,  however,  no  observation  upon  its  truth,  for  it  is  a  necessary  law 
of  the  relation  of  classes  in  extension.  It  is  an  immediate  consequence  of 
the  principle,  "For  every  y,  y  c  1",  that  is,  "All  members  of  any  class,  y, 
are  also  members  of  the  class  of  all  things".  One  cannot  accept  this  last 
without  accepting,  by  implication,  the  principle  that,  in  extension,  the  null- 
class  is  contained  in  every  class. 

The  interpretation  of  propositions  in  which  no  null-class  is  involved  is 
not  subject  to  any  corresponding  difficulty,  both  because  in  such  cases  the 
relations  predicated  are  frequently  thought  of  in  extension  and  because 
the  relation  of  classes  in  extension  is  entirely  analogous  to  their  relation  in 
intension  except  where  the  class  0  or  the  class  1  is  involved.  But  the 
interpretation  of  the  algebra  must,  in  all  cases,  be  confined  to  extension. 
In  brief:  "All  a  is  b"  must  always  be  interpreted  in  the  algebra  as  stating 
a  relation  of  classes  in  extension,  not  of  class-concepts,  and  this  requires 
that,  whenever  a  is  an  empty  class,  "All  a  is  6"  should  be  true. 

The  proposition,  "No  a  is  b",  is  represented  by  a  b  =  0 — "Nothing  is 
both  a  and  b",  or  "Those  things  which  are  members  of  a  and  of  b  both, 
do  not  exist".  Since  "No  a  is  6"  is  equivalent  to  "All  a  is  not-6",  it  may 
also  be  represented  by  a  -b  =  -b,  a  c  -b,  b  c  -a,  or  a  =  v  -b,  where  v  is 
undetermined.  In  the  case  of  this  proposition,  there  is  no  discrepancy 
between  the  algebra  and  the  ordinary  use  of  language. 

The  representation  of  particular  propositions  has  been  a  problem  to 
symbolic  logicians,  partly  because  they  have  not  clearly  conceived  the 


188  A  Survey  of  Symbolic  Logic 

relations  of  classes  and  have  tried  to  stretch  the  algebra  to  cover  traditional 
relations  which  hold  in  intension  only.  If  "Some  a  is  b"  be  so  interpreted 
that  it  is  false  when  the  class  a  has  no  members,  then  "Some  a  is  b"  will 
not  follow  from  "All  a  is  6",  for  "All  a  is  6"  is  true  whenever  a  =  0.  But 
on  the  other  hand,  if  "Some  a  is  6"  be  true  when  a  =  0,  we  have  two  diffi- 
culties: (1)  this  does  not  accord  with  ordinary  usage,  and  (2)  "Some  a  is  b" 
will  not,  in  that  case,  contradict  "No  a  is  b".  For  whenever  there  are  no 
members  of  a  (when  a  =  0),  "No  a  is  b"  (a  b  =  0)  will  be  true.  Hence  if 
"Some  a  is  6"  can  be  true  when  a  =  0,  then  "Some  a  is  b"  and  "No  a  is  b" 
can  both  be  true  at  once.  The  solution  of  the  difficulty  lies  in  observing 
that  "Some  a  is  6"  as  a  relation  of  extension  requires  that  there  be  some  a— 
that  at  least  one  member  of  the  class  a  exist.  Hence,  when  propositions 
are  interpreted  in  extension,  "Some  a  is  b"  does  not  follow  from  "All  a 
is  b",  precisely  because  whenever  a  =  0,  "All  a  is  b"  will  be  true.  But 
"Some  a  is  b"  does  follow  from  "All  a  is  b,  and  members  of  a  exist". 

To  interpret  properly  "Some  a  is  b",  we  need  only  remember  that  it 
is  the  contradictory  of  "No  a  is  b".  Since  "No  a  is  6"  is  interpreted  by 
a  b  =  0,  "Some  a  is  b"  will  be  a  b  =(=  0,  that  is,  "The  class  of  things  which 
are  members  of  a  and  of  b  both  is  not-null". 

It  is  surprising  what  blunders  have  been  committed  in  the  representation 
of  particular  propositions.  "Some  x  is  y"  has  been  symbolized  by  x  y  =  v, 
where  v  is  undetermined,  and  by  u  x  =  v  y,  where  u  and  v  are  undetermined. 
Both  of  these  are  incorrect,  and  for  the  same  reason :  An  "  undetermined  " 
element  may  have  the  value  0  or  the  value  1  or  any  other  value.  Conse- 
quently, both  these  equations  assert  precisely  nothing  at  all.  They  are 
both  of  them  true  a  priori,  true  of  every  x  and  y  and  in  all  cases.  For 
them  to  be  significant,  u  and  v  must  not  admit  the  value  0.  But  in  that 
case  they  are  equivalent  to  x  y  =|=  0,  which  is  much  simpler  and  obeys  well- 
defined  laws  which  are  consonant  with  its  meaning. 

Since  we  are  to  symbolize  "All  a  is  b"  by  a  -b  =  0,  it  is  clear  that  its 
contradictory,  "Some  a  is  not  b",  will  be  a  -b  =(=  0. 

To  sum  up,  then:  the  four  typical  propositions  will  be  symbolized  as 
follows : 

A.  All  a  is  b,         a-b  =  0. 

E.  No  a  is  b,        a  b  =  0. 

I.  Some  a  is  b,        a  b  4=  0. 

O.  Some  a  is  not  b,        a-b  ^  0. 

Each  of  these  four  has  various  equivalents : 12 
12 See  Chap,  u,  4-9  and  8-14. 


Applications  of  the  Boole-Schroder  Algebra  189 

A.  a  -b  =  0,  a  —  a  b,  -a+b  =  1,  -a  +  -b  =  -a,  a  c b,  and  -b  c -a  are 
all  equivalent. 

E.  a  b  =  0,  a  =  a  -b,  -a  +  -b  =  1,  -a  +  b  =  -a,  ac-b,  and  be -a  are 
all  equivalent. 

I.  a  b  +  0,  a  4=  fl  ~&>  -a  +  -b  =1=  1,  and  -a  +  b  =t=  -ct  are  all  equivalent. 

O.  a  -6  4=  0,  a  =1=  a  b,  -a  +  b  =f=  1,  and  -a  +  -b  =f=  -a  are  all  equivalent. 
The  reader  will  easily  translate  these  equivalent  forms  for  himself. 

With  these  symbolic  representations  of  A,  E,  I  and  O,  let  us  investigate 
the  relation  of  propositions  traditionally  referred  to  under  the  topics, 
"The  Square  of  Opposition",  and  "Immediate  Inference". 

That  the  traditional  relation  of  the  two  pairs  of  contradictories  holds, 
is  at  once  obvious.  If  a  -b  =  0  is  true,  then  a  -b  =1=  0  is  false;  if  a  -b  =  0 
is  false,  then  a  -b  =j=  0  is  true.  Similarly  for  the  pair,  a  b  =  0  and  a  b  4=  0. 

The  relation  of  contraries  is  defined:  Two  propositions  such  that  both 
may  be  false  but  both  cannot  be  true  are  "contraries".  This  relation  is 
traditionally  asserted  to  hold  between  A  and  E.  It  does  not  hold  in  ex- 
tension: it  fails  to  hold  in  the  algebra  precisely  whenever  the  subject  of 
the  two  propositions  is  a  null-class.  If  a  =  0,  then  a  -b  =  0  and  a  b  =  O.13 
That  is  to  say,  if  no  members  of  a  exist,  then  from  the  point  of  view  of 
extension,  "All  a  is  b"  and  "No  a  is  b"  are  both  true.  But  if  it  be  assumed 
or  stated  that  the  class  a  has  members  (a  =}=  0),  then  the  relation  holds. 

a  =  a  (b  +  -6)  =  a  b  +  a  -b. 

Hence  if  a  =f=  0,  then  a  b  +  a  -b  =1=  0. 

[8  •  17]  If  a  b  +  a  -6  4=  0  and  a  -b  =  0,  then  a  b  +  0.  (1) 

And  if  a  b  +  a  -b  4=  0  and  a  b  -  0,  then  a  -b  4=  0.  (2) 

We  may  read  the  last  two  lines : 

(1)  If  there  are  members  of  the  class  a  and  all  a  is  b,  then  "No  a  is  b" 
is  false. 

(2)  If  there  are  members  of  the  class  a  and  no  a  is  b,  then  "All  a  is  b" 
is  false. 

By  tradition,  the  particular  affirmative  should  follow  from  the  universal 
affirmative,  the  particular  negative  from  the  universal  negative.  As  has 
been  pointed  put,  this  relation  fails  to  hold  when  a  =  0.  But  it  holds  when- 
ever a  4=  0.  We  can  read  a  b  4=  0,  in  (1)  above,  as  "Some  a  is  b"  instead 
of  "'No  a  is  6'  is  false",  and  a  -b  4=  0,  in  (2),  as  "Some  a  is  not  6"  instead 
of  " '  All  a  is  b '  is  false  ".  We  then  have : 

13  See  Chap,  n,  1-5. 


190  A  Survey  of  Symbolic  Logic 

(1)  If  there  are  members  of  a,  and  all  a  is  b,  then  some  a  is  6. 

(2)  If  there  are  members  of  a,  and  no  a  is  6,  then  some  a  is  not  b. 

"  Subcontraries "  are  propositions  such  that  both  cannot  be  false  but 
both  may  be  true.  Traditionally  "Some  a  is  6"  and  "Some  a  is  not  b" 
are  subcontraries.  But  whenever  a  =  0,  a  b  =f=  0  and  a  -b  =j=  0  are  both 
false,  and  the  relation  fails  to  hold.  When  a  =f=  0,  it  holds.  Since  a  b  =  0 
is  "'Some  a  is  b'  is  false  ",  and  a  -b  =  0  is  "'Some  a  is  not  b'  is  false",  we 
can  read  (1)  and  (2)  above: 

(1)  If  there  are  members  of  a,  and  "Some  a  is  b"  is  false,  then  some  a 
is  not  b. 

(2)  If  there  are  members  of  a  and  "Some  a  is  not  6"  is  false,  then  some 
a  is  6. 

To  sum  up,  then:  the  traditional  relations  of  the  "  square  of  opposition " 
hold  in  the  algebra  whenever  the  subject  of  the  four  propositions  denotes  a 
class  which  has  members.  When  the  subject  denotes  a  null-class,  only 
the  relation  of  the  contradictories  holds.  The  two  universal  propositions 
are,  in  that  case,  both  true,  and  the  two  particular  propositions  both  false. 

The  subject  of  immediate  inference  is  not  so  well  crystallized  by  tradi- 
tion, and  for  the  good  reason  that  it  runs  against  this  very  difficulty  of  the 
class  without  members.  For  instance,  the  following  principles  would  all 
be  accepted  by  some  logicians: 

"No  a  is  b"  gives  "No  6  is  a". 

"No  b  is  a"  gives  "All  b  is  not-a". 

"All  b  is  not-a"  gives  "Some  b  is  not-a". 

"Some  b  is  not-a"  gives  "Some  not-a  is  b". 

Hence  "No  a  is  6"  gives  "Some  not-a  is  b". 

"No  cows  (a)  are  inflexed  gasteropods  (6)  "  implies  "Some  non-cows  are 
inflexed  gasteropods":  "No  mathematician  (a)  has  squared  the  circle  (6)" 
implies  "Some  non-mathematicians  have  squared  the  circle".  These  infer- 
ences are  invalid  precisely  because  the  class  b — inflexed  gasteropods,  suc- 
cessful circle-squarers — is  an  empty  class;  and  because  it  was  presumed 
that  "All  b  is  not-a"  gives  "Some  b  is  not-a".  Those  who  consider  the 
algebraic  treatment  of  null-classes  to  be  arbitrary  will  do  well  to  consider 
the  logical  situation  just  outlined  with  some  care.  The  inference  of  any 
particular  proposition  from  the  corresponding  universal  requires  the 
assumption  that  either  the  class  denoted  -by  the  subject  of  the  particular 
proposition  or  the  class  denoted  by  its  predicate  ("not-6"  regarded  as  the 
predicate  of  "Some  a  is  not  b")  is  a  class  which  has  members. 


Applications  of  the  Boole-Schroder  Algebra 


191 


The  "conversion  "  of  the  universal  negative  and  of  the  particular  affirma- 
tive is  validated  by  the  law  a  b  =  b  a.  "  No  a  is  b  ",  a  b  =  0,  gives  b  a  =  0, 
"No  b  is  a".  And  "Some  a  is  b",  a  b  +  0,  gives  b  a  4=  0,  "Some  b  is  a". 
Also,  "Some  a  is  not  b",  a  -b  =t=  0,  gives  -b  a  =j=  0,  "Some  not-6  is  a". 
The  "converse"  of  the  universal  affirmative  is  simply  the  "converse" 
of  the  corresponding  particular,  the  inference  of  which  from  the  universal 
has  already  been  discussed. 

What  are  called  "obverses" — i.  e.,  two  equivalent  propositions  with 
the  same  subject  and  such  that  the  predicate  of  one  is  the  negative  of  the 
predicate  of  the  other — are  merely  alternative  readings  of  the  same  equation, 
or  depend  upon  the  law,  -(-a)  =  a14.  Since  xy  =  0  is  " No  x  is  y  ",  a  -b  =  0, 
which  is  "All  a  is  b",  is  also  "No  a  is  not-6".  And  since  a  b  =  0  is  equiva- 
lent to  a  -(-b)  =  0,  "No  a  is  b"  is  equivalent  to  "All  a  is  not-6". 

A  convenient  diagram  for  immediate  inferences  can  be  made  by  putting 
S  (subject)  and  P  (predicate)  in  the  center  of  the  circles  assigned  to  them, 
-S  between  the  two  divisions  of  -S,  and  -P  between  its  two  constituent 
divisions.  The  eight  arrows  indicate  the  various  ways  in  which  the  dia- 


Gtiyen  Prop, 


Converse 


FIG.  13 

gram  may  be  read,  and  thus  suggest  all  the  immediate  inferences  which 
are  valid.  For  example,  the  arrow  marked  "converse"  indicates  the  two 
terms  which  will  appear  in  the  converse  of  the  given  proposition  and  the 
order  in  which  they  occur.  In  this  diagram,  we  must  specify  the  null  and 
"See  Chap,  n,  2-8. 


192 


A  Survey  of  Symbolic  Logic 


not-null  regions  indicated  by  the  given  proposition.     And  we  may  —  if  we 
wish  —  add  the  qualification  that  the  classes,  S  and  P,  have  members. 
If  "No  S  is  P",  and  S  and  P  have  members: 


SP  =  0, 


0 


5. 


-S-P 


FIG.  14 


Reading  the  diagram  of  figure  14  in  the  various  possible  ways,  we  have: 

1.  No  S  is  P,     and  1.  Some  S  is  not  P.     (According  as  we  read  what 
is  indicated  by  the  fact  that  S  P  is  null,  or  what  is  indicated  by  the  fact 
that  S  -P  is  not-null.) 

2.  All  S  is  not-P,     and  2.  Some  S  is  not-P. 

3.  All  P  is  not-S,    and  3.  Some  P  is  not-S. 

4.  No  P  is  S,     and  4.  Some  P  is  not-S. 

5.  Wanting. 

6.  Some  not-S  is  P. 

7.  Some  not-P  is  S. 

8.  Wanting. 


Applications  of  the  Boole-Schroder  Algebra 


193 


Similarly,  if  "All  S  is  P",  and  S  and  P  have  members: 
S-P=.Q,        S*Q,        P  +  0 
S.  3. 


A 


.6* 


FIG.  15 


Reading  from  the  diagram  (figure  15),  we  have: 

1.  All  S  is  P,     and  1.  Some  S  is  P. 

2.  No  S  is  not-P. 

3.  Wanting. 

4.  Some  P  is  S. 

5.  Some  not-S  is  not-P. 

6.  Wanting. 

7.  No  not-P  is  S. 

8.  All  not-P  is  not-S,    and  8.  Some  not-P  is  not-S. 

The  whole  subject  of  immediate  inference  is  so  simple  as  to  be  almost 
trivial.  Yet  in  the  clearing  of  certain  difficulties  concerning  null-classes 
the  algebra  has  done  a  real  service  here. 

The  algebraic  processes  which  give  the  results  of  syllogistic  reasoning 
have  already  been  illustrated.  But  in  those  examples  we  carried  out  the 
14 


194  A  Survey  of  Symbolic  Logic 

operations  at  unnecessary  lengths  in  order  to  illustrate  their  connection 
with  the  diagrams.  The  premises  of  any  syllogism  give  information  which 
concerns,  altogether,  three  classes.  The  object  is  to  draw  a  conclusion 
which  gives  as  much  of  this  information  as  can  be  stated  independently  of 
the  "middle"  term.  This  is  exactly  the  kind  of  result  which  elimination 
gives  in  the  algebra.  And  elimination  is  very  simple.  The  result  of 
eliminating  x  from  A  x  +  B  -x  =  0  is  A  B  =  O.15  Whenever  the  conclusion 
of  a  syllogism  is  universal,  it  may  be  obtained  by  combining  the  premises 
in  a  single  equation  one  member  of  which  is  0,  and  eliminating  the  "middle" 
term.  For  example : 

No  x  is  y,         x  y  =  0. 

All  z  is  x,        z-x  =  0. 
Combining  these,  x  y  +  z  -x  =  0. 
Eliminating  x,  y  z  =  0. 

Hence  the  valid  conclusion  is  "No  y  is  z",  or  "No  z  is  y". 

Any  syllogism  with  a  universal  conclusion  may  also  be  symbolized  so 
that  the  conclusion  follows  from  the  law,  "If  a  cb  and  b  cc,  then  ace". 
By  this  method,  the  laws,  -(-a)  =  a  and  "If  a  cb,  then  -b  c-a",  are  some- 
times required  also.16  For  example: 

No  x  is  y,        x  c  -y. 
All  z  is  x,        z  ex. 
Hence  z  c-y,  or  "No  z  is  y",  and  y  c-z,  or  "No  y  is  z". 

There  is  no  need  to  treat  further  examples  of  syllogisms  with  universal 
conclusions:  they  are  all  alike,  as  far  as  the  algebra  is  concerned.  Of  course, 
there  are  other  ways  of  representing  the  premises  and  of  getting  the  con- 
clusion, but  the  above  are  the  simplest. 

When  a  syllogism  has  a  particular  premise,  and  therefore  a  particular 
conclusion,  the  process  is  somewhat  different.  Here  we  have  given  one 
equation  {  =  0 }  and  one  inequation  { =f=  0 } .  WTe  proceed  as  follows : 
(1)  expand  the  inequation  by  introducing  the  third  element;  (2)  multiply 
the  equation  by  the  element  not  appearing  in  it;  (3)  make  use  of  the  prin- 
ciple, "  If  a  +  b  4=  0  and  a  =  0,  then  6  4=  0",  to  obtain  an  inequation  with 
only  one  term  in  the  literal  member;  (4)  eliminate  the  element  representing 
the  "middle  term"  from  this  inequation.  Take,  for  example,  A  1 1  in 

15  See  Chap,  n,  7-4. 

"See  Chap,  n,  2-8  and  3-1. 


Applications  of  the  Boole-Schroder  Algebra  195 

the  third  figure: 

All  x  is  z,         x  -z  =  0. 

Some  x  is  y,         x  y  =f=  0. 

x  y  =  x  y  (z  +  -z)  =  x  y  z  +  xy  -z.     Hence,  x  y  z  +  xy  -z  4=  0. 
[1  •  5]  Since  x  -z  =  0,  x  y  -z  =  0. 

[8  •  17]  Since  xy  z  +  xy  -z  =f=  0  and  x  y  -x  =  0,  therefore  x  y  z  ={=  0. 
Hence  [8-221  yz  +  0,  or  "Some  y  is  z". 

An  exactly  similar  process  gives  the  conclusion  for  every  syllogism  with  a 
particular  premise. 

We  have  omitted,  so  far,  any  consideration  of  syllogisms  with  both 
premises  universal  and  a  particular  conclusion — those  with  "weakened" 
conclusions,  and  A  A  I  and  E  A  0  in  the  third  and  fourth  figures.  These 
are  all  invalid  as  general  forms  of  reasoning.  They  involve  the  difficulty 
which  is  now  familiar:  a  universal  does  not  give  a  particular  without  an 
added  assumption  that  some  class  has  members.  If  we  add  to  the  premises 
of  such  syllogisms  the  assumption  that  the  class  denoted  by  the  middle 
term  is  a  class  with  members,  this  makes  the  conclusion  valid.  Take,  for 
example,  A  A  I  in  the  third  figure : 

All  x  is  y,        x-y  =  0,         and  x  has  members,        x  =t=  0. 
All  x  is  z,         x-z  =  0. 

Since  x  4=  0,  x  y  +  x  -y  4=  0,  and  since  x  -y  =  0,  x  y  4=  0. 
Hence  x  y  z  +  x  y  -z  4=  0.  (1) 

Since  x  -z  =  0,  x  y  -z  =  0.  (2) 

By  (1)  and  (2),  x  y  z  +  0,  and  hence  y  z  =(=  0,  or  "Some  y  is  z". 

Syllogisms  of  this  form  are  generally  considered  valid  because  of  a  tacit 
assumption  that  we  are  dealing  with  things  which  exist.  In  symbolic 
reasoning,  or  any  other  which  is  rigorous,  any  such  assumption  must  be 
made  explicit. 

An  alternative  treatment  of  the  syllogism  is  due  to  Mrs.  Ladd-Franklin.17 
If  we  take  the  two  premises  of  any  syllogism  and  the  contradictory  of  its 
conclusion,  we  have  what  may  be  called  an  "inconsistent  triad" — three 
propositions  such  that  if  any  two  of  them  be  true,  the  third  must  be  false. 
For  if  the  two  premises  be  true,  the  conclusion  must  be  true  and  its  con- 

17  See  "On  the  Algebra  of  Logic",  in  Studies  in  Logic  by  members  of  Johns  Hopkins 
University,  ed.  by  Peirce;  also  articles  listed  in  Bibl.  We  do  not  follow  Mrs.  Franklin's 
symbolism  but  give  her  theory  in  a  modified  form,  due  to  Josiah  Royce. 


196  A  Survey  of  Symbolic  Logic 

tradictory  false.  And  if  the  contradictory  of  the  conclusion  be  true,  i.  e., 
if  the  conclusion  be  false,  and  either  of  the  premises  true,  then  the  other 
premise  must  be  false.  As  a  consequence,  every  inconsistent  triad  corre- 
sponds to  three  valid  syllogisms.  Any  two  members  of  the  triad  give  the 
contradictory  of  the  third  as  a  conclusion.  For  example: 

Inconsistent  Triad 

1.  All  x  is  y 

2.  Ally  is  z 

3.  Some  x  is  not  z. 

Valid  Syllogisms 

1.  All  x  is  y  1.  All  x  is  y  2.  All  y  is  2 

2.  All  y  is  z                      3.  Some  x  is  not  z  3.  Some  x  is  not  2 
.'.  All  x  is  z.                      .'.  Some  y  is  not  z.  .'.  Some  x  is  not  y. 

Omitting  the  cases  in  which  two  universal  premises  are  supposed  to 
give  a  particular  conclusion,  since  these  really  have  three  premises  and 
are  not  syllogisms,  the  inconsistent  triad  formed  from  any  valid  syllogism 
will  consist  of  two  universals  and  one  particular.  For  two  universals  will 
give  a  universal  conclusion,  whose  contradictory  will  be  a  particular;  while 
if  one  premise  be  particular,  the  conclusion  will  be  particular,  and  its 
contradictory  wrill  be  the  second  universal.  Representing  universals  and 
particulars  as  we  have  done,  this  means  that  if  we  symbolize  any  incon- 
sistent triad,  we  shall  have  two  equations  {  =  0}  and  one  inequation  {  4=  0}. 
And  the  two  universals  {  =  0 }  must  give  the  contradictory  of  the  particular 
as  a  conclusion.  This  means  that  the  contradictory  of  the  particular 
must  be  expressible  as  the  elimination  resultant  of  an  equation  of  the  form 
ax  +  b-x  =  0,  because  we  have  found  all  conclusions  from  two  universals 
to  be  thus  obtainable.  Hence  the  two  universals  of  any  inconsistent  triad 
will  be  of  the  form  a  x  =  0  and  b  -x  —  0  respectively.  The  elimination 
resultant  of  a  x  +  b  -x  =  0  is  a  b  =  0,  whose  contradictory  will  be  a  b  4=  0. 
Hence  every  inconsistent  triad  will  have  the  form : 

ax  =  0,         b-x  =  0,         a  6  4=  0 

where  a  and  b  are  any  terms  whatever  positive  or  negative,  and  x  is  any 
positive  term. 

The  validity  of  any  syllogism  may  be  tested  by  expressing  its  proposi- 
tions in  the  form  suggested,  contradicting  its  conclusion  by  changing  it 
from  {  =  0}  to  { =£  0}  or  the  reverse,  and  comparing  the  resulting  triad 


Applications  of  the  Boole-Schroder  Algebra  197 

with  the  above  form.  And  the  conclusion  of  any  syllogism  may  be  got  by 
considering  how  the  triad  must  be  completed  to  have  the  required  form. 
Thus,  if  the  two  premises  are 

No  x  is  y,  x  y  =  0 

and        All  not-2  is  y,        -z  -y  =  0 

the  conclusion  must  be  universal.  The  particular  required  to  complete 
the  triad  is  x  -z  4=  0.  Hence  the  conclusion  is  x  -z  =  0,  pr  "All  x  is  2". 
(Incidentally  it  may  be  remarked  that  this  valid  syllogism  is  in  no  one  of 
the  Aristotelian  moods.)  Again,  if  the  premises  should  be  x  y  =  0  and 
y  z  =  0,  no  conclusion  is  possible,  because  these  two  cannot  belong  to  the 
same  inconsistent  triad. 

We  can,  then,  frame  a  single  canon  for  all  strictly  valid  syllogistic  reason- 
ing: The  premises  and  the  contradictory  of  the  conclusion,  expressed  in 
symbolic  form,  {  =  0 }  or  {  4=  0 } ,  must  form  a  triad  such  that 

(1)  There  are  two  universals  {  =  0}  and  one  particular  {  4=  0}. 

(2)  The  two  universals  have  a  term  in  common,  which  is  once  positive 
and  once  negative. 

(3)  The  particular  puts  =(=  0  the  product  of  the  coefficients  of  the  com- 
mon term  in  the  two  universals. 

A  few  experiments  with  traditional  syllogisms  will  make  this  matter  clear 
to  the  reader.  The  validity  of  this  canon  depends  solely  upon  the  nature 
of  the  syllogism — three  terms,  three  propositions — and  upon  the  law  of 
elimination  resultants,  "If  a  x  +  b  -x  =  0,  then  a  b  =  0". 

Reasoning  which  involves  conditional  propositions — hypothetical  argu- 
ments, dilemmas,  etc. — may  be  treated  by  the  same  process,  if  we  first 
reduce  them  to  syllogistic  form.  For  example,  we  may  translate  "If  A 
is  B,  then  C  is  D"  by  "All  x  is  y",  where  x  is  the  class  of  cases  in  which 
A  is  B,  and  y  the  class  of  cases  in  which  C  is  D — i.  e.,  "All  cases  in  which  A 
is  B  are  cases  in  which  C  is  D".  And  we  may  translate  "But  A  is  B" 
by  "  All  z  is  x  ",  where  2  is  the  case  or  class  of  cases  under  discussion.  Thus 
the  hypothetical  argument:  "If  A  is  B,  C  is  D.  But  A  is  B.  Therefore, 
C  is  D",  is  represented  by  the  syllogism: 

"All  cases  in  which  A  is  B  are  cases  in  which  C  is  Z). 

"  But  all  the  cases  in  question  are  cases  in  which  A  is  B. 

"Hence  all  the  cases  in  question  are  cases  in  which  C  is  D." 
And  all  other  arguments  of  this  type  are  reducible  to  syllogisms  in  some 
similar  fashion.     Thus  the  symbolic  treatment  of  the  syllogism  extends  to 


198  A  Survey  of  Symbolic  Logic 

them  also.  But  conditional  reasoning  is  more  easily  and  simply  treated 
by  another  interpretation  of  the  algebra — the  interpretation  for  propositions. 

The  chief  value  of  the  algebra,  as  an  instrument  of  reasoning,  lies  in 
its  liberating  us  from  the  limitation  to  syllogisms,  hypothetical  arguments, 
dilemmas,  and  the  other  modes  of  traditional  logic.  Many  who  object  to 
the  narrowness  of  formal  logic  still  do  not  realize  how  arbitrary  (from  the 
logical  point  of  view)  its  limitations  are.  The  reasons  for  the  syllogism, 
etc.,  are  not  logical  but  psychological.  It  may  be  worth  while  to  exemplify 
this  fact.  We  shall  offer  two  illustrations  designed  to  show,  each  in  a 
different  way,  a  wide  range  of  logical  possibilities  undreamt  of  in  formal 
logic.  The  first  of  these  turns  upon  the  properties  of  a  triadic  relation 
whose  significance  was  first  pointed  out  by  Mr.  A.  B.  Kempe.18 

It  is  characteristically  human  to  think  in  terms  of  dyadic  relations: 
we  habitually  break  up  a  triadic  relation  into  a  pair  of  dyads.  In  fact,  so 
ingrained  is  this  disposition  that  some  will  be  sure  to  object  that  a  triadic 
relation  is  a  pair  of  dyads.  It  would  be  exactly  as  logical  to  maintain  that 
all  dyadic  relations  are  triads  with  a  null  member.  Either  statement  is 
correct  enough :  the  difference  is  simply  one  of  point  of  view — psychological 
preference.  If  there  should  be  inhabitants  of  Mars  whose  logical  sense 
coincided  with  our  own,  so  that  any  conclusion  which  seemed  valid  to  us 
would  seem  valid  to  them,  and  vice  versa,  but  whose  psychology  otherwise 
differed  from  ours,  these  Martians  might  have  an  equally  fundamental 
prejudice  in  favor  of  triadic  relations.  We  can  point  out  one  such  which 
they  might  regard  as  the  elementary  relation  of  logic — as  we  regard  equality 
or  inclusion.  In  terms  of  this  triadic  relation,  all  their  reasoning  might 
be  carried  out  with  complete  success. 

Let  us  symbolize  by  (ac/b),  a  -b  c  +  -a  b  -c  =  0.  This  relation  may  be 
diagrammed  as  in  figure  16,  since  a  -b  c  +  -a  b  -c  =  0  is  equivalent  to 
a  c  c  b  c  (a  +  c) .  (Note  that  (ac/b)  and  (ca/b)  are  equivalent,  since  a -be 
•f  -a  b  -c  is  symmetrical  with  respect  to  a  and  c.) 

This  relation  (ac/b)  represents  precisely  the  information  which  we 
habitually  discard  in  drawing  a  syllogistic  conclusion  from  two  universal 
premises.  If  all  a  is  b  and  all  b  is  c,  we  have 

a  -b  =  0        and         b  -c  =  0 

Hence  a  -b  (c  +  -c)  +  (a  +  -a)  b  -c  =  0, 

18  See  his  paper  "On  the  Relation  of  the  Logical  Theory  of  Classes  and  the  Geometrical 
Theory  of  Points,",  Proc.  London  Math.  Soc.,  xxi,  147-82.  But  the  use  we  here  make  of 
this  relation  is  due  to  Josiah  Royce.  For  a  further  discussion  of  Kempe's  triadic  relation, 
see  below,  Chap,  vi,  Sect.  iv. 


Applications  of  the  Boole-Schroder  Algebra 


199 


Or,  a  -b  c  +  a  -b  -c  +  a  b  -c  +  -a  b  -c  =  0. 

[5  •  72]  This  equation  is  equivalent  to  the  pair, 

(1)  a-b-c  +  ab  -c  =  a  -c  (b  +  -b)  =  a  -c  =  0, 
and  (2)  a  -b  c  +  -a  b  -c  =  0. 

(1)  is  the  syllogistic  conclusion,  "All  a  is  c";  (2)  is  (ac/6).  Perhaps  most 
of  us  would  feel  that  a  syllogistic  conclusion  states  all  the  information 
given  by  the  premises:  the  Martians  might  equally  well  feel  that  precisely 


FIG.  16 


what  we  overlook  is  the  only  thing  worth  mentioning.  And  yet  with  this 
curious  "illogical"  prejudice,  they  would  still  be  capable  of  understanding 
and  of  getting  for  themselves  any  conclusion  which  a  syllogism  or  a  hypo- 
thetical argument  can  give,  and  many  others  which  are  only  very  awkwardly 
stateable  in  terms  of  our  formal  logic.  Our  relation,  a  cb,  or  "All  a  is  b", 
would  be,  in  their  terms,  (06/a).  (Ofe/a)  is  equivalent  to 

l-a-b  +  0--ab  =  0  =  a  -b 

Hence  the  syllogism  in  Barbara  would  be  "  (Ob/a)  and  (Oc/6),  hence  (Oc/a) ". 
This  would,  in  fact,  be  only  a  special  case  of  a  more  general  principle  which 
is  one  of  those  we  may  suppose  the  Martians  would  ordinarily  rely  upon 
for  inference:  "If  (xb/a)  and  (xc/b),  then  (xc/a)".  That  this  general 
principle  holds,  is  proved  as  follows: 

(xbja}  is  "X  a  -b  +  x  -a  b  =  0 
(xc/b)  is   -x  b  -c  +  x  -b  c  =  0 
These  two  together  give: 

-x  a-b  (c  +  -c)  +  x  -a  b  (c  +  -c)  +  -x  b  -c  (a  +  -a)  +  x  -b  c  (a  +  -a)  =  0, 
or,  -x  a  -b  c  +  -x  a  -b  -c  +  x  -a  b  c  +  x  -a  b  -c  +  -x  a  b  -c  +  -x  -a  b  -c 

+  xa-b  c  +  x-a-bc  =  0. 


200  A  Survey  of  Symbolic  Logic 

[5-72]  This  equation  is  equivalent  to  the  pair, 

(1)  ~x  a  b  -c  +  -x  a  -b  -c  +  x  -a  b  c  +  x  -a  -b  c 

=  -x  a-c  (b  +  -b)  +  x  -a  c  (b  +  -6) 
=  -x  a  -c  +  x  -a  c  =  0. 

(2)  x  -a  b  -c  +  -x  -a  b  -c  +  x  a  -b  c  +  -x  a  -b  c 

=  -a  b  -c  (x  +  -x)  +  a  -b  c  (x  +  -x) 
=  -a  b  -c  +  a  -b  c  =0. 

(1)  is  (xc/a),  of  which  our  syllogistic  conclusion  is  a  special  case;  (2)  is  a 
similar  valid  conclusion,  though  one  which  we  never  draw  and  have  no 
language  to  express. 

Thus  these  Martians  could  deal  with  and  understand  our  formal  logic 
by  treating  our  dyads  as  triads  with  one  member  null.  In  somewhat 
similar  fashion,  hypothetical  propositions,  the  relation  of  equality,  syllo- 
gisms with  a  particular  premise,  dilemmas,  etc.,  are  all  capable  of  state- 
ment in  terms  of  the  relation  (acjb).  As  a  fact,  this  relation  is  much  more 
powerful  than  any  dyad  for  purposes  of  reasoning.  Anyone  who  will 
trouble  to  study  its  properties  will  be  convinced  that  the  only  sound  reason 
for  not  using  it,  instead  of  our  dyads,  is  the  psychological  difficulty  of 
keeping  in  mind  at  once  two  triads  with  two  members  in  common  but 
differently  placed,  and  a  third  member  which  is  different  in  the  two.  Our 
attention-span  is  too  small.  But  the  operations  of  the  algebra  are  inde- 
pendent of  such  purely  psychological  limitations — that  is  to  say,  a  process 
too  complicated  for  us  in  any  other  form  becomes  sufficiently  simple  to  be 
clear  in  the  algebra.  The  algebra  has  a  generality  and  scope  which  "  formal " 
logic  cannot  attain. 

This  illustration  has  indicated  the  possibility  of  entirely  valid  non- 
traditional  modes  of  reasoning.  We  shall  now  exemplify  the  fact  that  by 
modes  wilich  are  not  so  remote  from  familiar  processes  of  reasoning,  any 
number  of  non-traditional  conclusions  can  be  drawn.  For  this  purpose, 
we  make  use  of  Poretsky's  Law  of  Forms:19 

x  =  0  is  equivalent  to  t  =  t  -x  +  -t  x 

This  law  is  evident  enough:  if  x  —  0,  then  for  any  t,  t-x  =  M  =  t,  and 
-tx  =  -t-0  =  0,  while  t+  0  =  t.  Let  us  now  take  the  syllogistic  premises, 
"All  a  is  6"  and  "All  b  is  c",  and  see  what  sort  of  results  can  be  derived 
from  them  by  this  law. 

All  a  is  b,        a-b  =  0. 
All  b  is  c,         b-c  =  0. 
19  See  Chap,  n,  7-15  and  7-16. 


Applications  of  the  Boole-Schroder  Algebra  201 

Combining  these,  a-b  +  b  -c  =  0. 

And  [3-4-41]  -(a-b  +  b-c}  =  -(a -&)•-(&  -c)  =  (-a  +  6) (-6  +  c) 

=  -a  -6  +  -a  c  +  b  c. 
Let  us  make  substitutions,  in  terms  of  a,  b,  and  c,  for  the  t  of  this  formula. 

a  +  b  =  (a  +  6)  (-a  -6  +  -a  c  +  b  c)  +  -a  -6  (a-b  +  b  -c) 
=  abc  +  -abc  +  bc  =  6c 

What  is  either  a  or  6  is  identical  with  that  which  is  both  6  and  c.  This  is  a 
non-syllogistic  conclusion  from  "All  a  is  b  and  all  b  is  c".  Other  such 
conclusions  may  be  got  by  similar  substitutions  in  the  formula. 

a  +  c  =  (a  +  c)  (-a  -6  +  -a  c  +  b  c)  +  -a  -c  (a-b  +  b  -c) 

=  a  6  c  +  -a  -&  c  +  -a  c  +  b  c  +  -a  b  -c  =  a  b  c  +  -a  (b  +  c) . 

What  is  either  a  or  c  is  identical  with  that  which  is  a,  b,  and  c,  all  three,  or 
is  not  a  and  either  b  or  c. 

-6  c  =  -b  c  (-a -b  +  -a c  +  b  c)  +  (b  +  -c) (a-b  +  b  -c) 
=  -a  -b  c  +  b  -c  +  a  -b  -c  =  -a  -b  c  +  (a  +  6)  -c 

That  which  is  b  but  not  c  is  identical  with  what  is  c  but  neither  a  nor  b 
or  is  either  a  or  6  but  not  c.  The  number  of  such  conclusions  to  be  got  from 
the  premises,  "All  a  is  b"  and  "All  b  is  c",  is  limited  only  by  the  number  of 
functions  which  can  be  formed  with  a,  b,  and  c,  and  the  limitation  to  sub- 
stitutions in  terms  of  these  is,  of  course,  arbitrary.  By  this  method,  the 
number  of  conclusions  which  can  be  drawn  from  given  premises  is  entirely 
unlimited. 

In  concluding  this  discussion  of  the  application  of  the  algebra  to  the 
logic  of  classes,  we  may  give  a  few  examples  in  which  problems  more  involved 
than  those  usually  dealt  with  by  formal  logic  are  solved.  The  examples 
chosen  are  mostly  taken  from  other  sources,  and  some  of  them,  like  the 
first,  are  fairly  historic. 

Example  I.20 

A  certain  club  has -the  following  rules:  (a)  The  financial  committee 
shall  be  chosen  from  among  the  general  committee;  (6)  No  one  shall  be  a 
member  both  of  the  general  and  library  committees  unless  he  be  also  on 
the  financial  committee;  (c)  No  member  of  the  library  committee  shall  be 
on  the  financial  committee. 

Simplify  the  rules. 

10  See  Venn,  Symbolic  Logic,  ed.  2,  p.  331. 


202  A  Survey  of  Symbolic  Logic 

Let  /  =  member  of  financial  committee. 
g=        "        "  general 
1=        "        "  library  "        . 

The  premises  then  become: 
(a)  fcg,        or        f-g  =  0. 

,        or        -l  =  0. 


(c)  /  /  =  0. 

We  can  discover  by  diagramming  whether  there  is  redundancy  here.     In 
figure  17,  (a)  is  indicated  by  vertical  lines,  (6)  by  horizontal,  (c)  by  oblique. 


(a)  and  (c)  both  predicate  the  non-existence  of  /  -g  I.     To  simplify  the 
rules,  unite  (a),  (6),  and  (c)  in  a  single  equation: 


Hence,  /  -g  +  -f  g  I  +/  /  (g  +  -g)  =  f  -g  +  -f  g  I  +f  g  I  +f  -g  I 

[5-91]  =  f-g+(-f+f)gl=f-g  +  gl   =   Q. 

And  [5-72]  this  is  equivalent  to  the  pair,  f-g  =  0  and  g  I  =  0. 

Thus  the  simplified  rules  will  be : 

(«')  The  financial  committee  shall  be  chosen  from  among  the  general 
committee. 

(6')  No  member  of  the  general  committee  shall  be  on  the  library  com- 
mittee. 


Applications  of  the  Boole-Schroder  Algebra 


203 


Example  2.21 

The  members  of  a  certain  collection  are  classified  in  three  ways — as 
a's  or  not,  as  6's  or  not,  and  as  c's  or  not.  It  is  then  found  that  the  class  b 
is  made  up  precisely  of  the  a's  which  are  not  c's  and  the  c's  which  are  not  a's. 
How  is  the  class  c  constituted? 

Given :  b  =  a  -c  +  -a  c.    To  solve  for  c.22 

b  =  b  (c  +  -c)  =  bc  +  b-c. 
Hence,  bc  +  b-c  =  a  -c  +  -a  c. 
Hence  [7-27]  a  -6  +  -a  6  c  c  c  a  -b  +  -a  b. 
Or  [2-2]  c  =  a -6  + -a  6. 

The  c's  comprise  the  a's  which  are  not  6's  and  the  6's  which  are  not  a's. 

Another  solution  of  this  problem  would  be  given  by  reducing  6  =  a  -c 
+  -ac  to  the  form  {  =  0}  and  using  the  diagram. 
[7  •  1]  6  =  a  -c  +  -a  c  is  equivalent  to 

6  •  -(a  -c  +  -a  c)  +  -6  (a  -c  +  -a  c)  =0 

And  [6  •  4]  -(a  -c  +  -a  c)  =  a  c  +  -a  -c. 
Hence,  a  6  c  +  -a  6  -c  +  a  -6  -c  +  -a  -6  c  =  0. 

We  observe  here  (figure  18)  not  only  that  c  =  a  -6  +  -a  6,  but  that  the 


FIG.  18 

relation  of  a,  6,  and  c,  stated  by  the  premise  is  totally  symmetrical,  so  that 
we  have  also  a  =  6  -c  +  -6  c. 

21  Adapted  from  one  of  Venn's,  first  printed  in  an  article  on  "Boole's  System  of  Logic", 
Mind,  i  (1876),  p.  487. 

22  This  proof  will  be  intelligible  if  the  reader  understands  the  solution  formula  referred  to. 


204 


A  Survey  of  Symbolic  Logic 


Example  3.M 

If  x  that  is  not  a  is  the  same  as  6,  and  a  that  is  not  x  is  the  same  as  c, 
what  is  x  in  terms  of  a,  b,  and  c? 

Given:  6  =  -ax  and  c  =  a  -.r.     To  solve  for  x. 

[7-1]  b  =  -ax  is  equivalent  to 

-(-a  x)  b  +  -a  -b  x  =  0  =  (a  +  -a:)  b  +  -a  -b  x 

=  ab  +  b-x  +  -a-bx  =  Q     (1) 
And  c  =  a  -x  is  equivalent  to 

-(a  -x)c  +  a  -c  -x  =  0  =  (-a  +  x)  c  +  a  -c  -x 


Combining  (1)  and  (2), 

a  b  +  -a  c  +  (-a  -b  +  c)  x  +  (b  +  a  -c)  -x  =  0 
Hence  [5-72]  (-a  -b  +  c)  x  +  (b  +  a  -c)  -x  =  0 

[7-221]  This  gives  the  equation  of  condition, 

(-a  -b  +  c)(b  +  a-c)  =  b  c  =  0 
[7-2]  The  solution  of  (4)  is 

(b  +  a-c)  ex  c-(-a  -b  +  c) 
And  by  (5), 

-(-a  -b  +  c)  =  -(-a  -b  +  c) 


-c-x  =  Q     (2) 


(3) 
(4) 

(5) 


Hence  [2-2]  x  =  b  +  a  -c. 


=  (a  +  6)  -c  +  6  c 

=  a  -c  +  6  (c  +  -c)  =  b  +  a  -c 


FIG.  19 
u  See  Lambert,  Logische  Abhandlungen,  i,  14. 


Applications  of  the  Boole-Schroder  Algebra 


205 


This  solution  is  verified  by  the  diagram  (figure  19)  of  equation  (3),  which 
combines  all  the  data.     Lambert  gives  the  solution  as 

x  =  (a  +  6)  -c 
This  also  is  verified  by  the  diagram. 

Example  4.24 

What  is  the  precise  point  at  issue  between  two  disputants,  one  of  whom, 
A,  asserts  that  space  should  be  defined  as  three-way  spread  having  points 
as  elements,  while  the  other,  B,  insists  that  space  should  be  defined  as 
three-way  spread,  and  admits  that  space  has  points  as  elements. 

Let  s  =  space, 

t  =  three-way  spread, 
p  =  having  points  as  elements. 
A  asserts :  s  =  t  p.        B  states :  s  =  t  and  s  c  p. 
s  =  t  p  is  equivalent  to 

s--(tp)+-stp  =  Q  =  s-t  +  s-p  +  -stp  =  Q  (1) 

s  c  p  is  equivalent  to  s  -p  =  0  (2) 

And  s  =  t  is  equivalent  to  s  -t  +  -s  t  =  0  (3) 
(2)  and  (3)  together  are  equivalent  to 

s  -t  +  s  -p  +  -s  t  =  0  (4) 

(1)  represents  ^4's  assertion,  and  (4)  represents  5's.     The  difference  between 


FIG.  20 


the  two  is  that  between  -s  t  p  =  0  and  -st  =  0.     (See  figure  20.) 

-st  =  -s  tp  +  -st-p 
24  Quoted  from  Jevons  by  Mrs.  Ladd-Franklin,  loc.  tit.,  p.  52. 


206  A  Survey  of  Symbolic  Logic 

The  difference  is,  then,  that  B  asserts  -st-p  =  0,  while  A  does  not.  It 
would  be  easy  to  misinterpret  this  issue,  -s  t-p  =  0  is  t-p  cs,  "Three- 
way  spread  not  having  points  as  elements,  is  space  ".  But  B  cannot  sig- 
nificantly assert  this,  for  he  has  denied  the  existence  of  any  space  not  having 
points  as  elements.  Both  assert  s  =  tp.  The  real  difference  is  this:  B 
definitely  asserts  that  all  three-way  spread  has  points  as  elements  and  is 
space,  wrhile  A  has  left  open  the  possibility  that  there  should  be  three-way 

spread  not  having  points  as  elements  which  should  not  be  space. 

*> 

Example  5. 

Amongst  the  objects  in  a  small  boy's  pocket  are  some  bits  of  metal 
which  he  regards  as  useful.  But  all  the  bits  of  metal  which  are  not  heavy 
enough  to  sink  a  fishline  are  bent.  And  he  considers  no  bent  object  useful 
unless  it  is  either  heavy  enough  to  sink  a  fishline  or  is  not  metal.  And  the 
only  objects  heavy  enough  to  sink  a  fishline,  which  he  regards  as  useful, 
are  bits  of  metal  that  are  bent.  Specifically  what  has  he  in  his  pocket  which 
he  regards  as  useful? 

Let  x  =  bits  of  metal, 

y  =  objects  he  regards  as  useful, 

z  =  things  heavy  enough  to  sink  a  fishline, 

w  =  bent  objects. 

Symbolizing  the  propositions  in  the  order  stated,  we  have 

xy*0 

x-zc.w,        or  x  -z  -w  =  0 

y  w  c  (z  +  -x),  or        xy-zw  =  0 

zy  c.x  w,        or  -x  y  z  +  y  z  -w  =  0 

Expanding  the  inequation  with  reference  to  z  and  w, 

xyzw  +  xyz-w  +  xy-zw  +  xy-z-w  4=  0 
Combining  the  equations, 

• 

x  -z  -w  (y  +  -y~)  +  x  y  -z  w  +  -x  y  z  (w  +  —w}  +  y  z-w  (x  +  -x)  =  0 
or       xy  -z-w  +  x-y  -z-w  +  x  y  -zw  +  -x  y  zw  +  -x  y  z-w  +  x  y  z-w  =  0 

All  the  terms  of  the  inequation  appear  also  in  this  equation,  with  the 
exception  of  x  y  z  w.  Hence,  by  8-17,  x  y  z  w  4=  0.  The  small  boy  has 


Applications  of  the  Boole-Schroder  Algebra 


207 


some  bent  bits  of  metal  heavy  enough  to  sink  a  fishline,  which  he  considers 
useful.     This  appears  in  the  diagram  (figure  21)  by  the  fact  that  while 


FIG.  21 

some  subdivision  of  x  y  must  be  not-null,  all  of  these  but  x  y  z  w  is  null. 
It  appears  also  that  anything  else  he  may  have  which  he  considers  useful 
may  or  may  not  be  bent  but  is  not  metal. 

Example  6.25 

The  annelida  consist  of  all  invertebrate  animals  having  red  blood  in  a 
double  system  of  circulating  vessels.  And  all  annelida  are  soft-bodied, 
and  either  naked  or  enclosed  in  a  tube.  Suppose  we  wish  to  obtain  the 
relation  in  which  soft-bodied  animals  enclosed  in  tubes  are  placed  (by  virtue 
of  the  premises)  with  respect  to  the  possession  of  red  blood,  of  an  external 
covering,  and  of  a  vertebral  column. 

Let  a  =  annelida, 

s  =  soft-bodied  animals, 
n  =  naked, 

t  =  enclosed  in  a  tube, 
i  =  invertebrate, 
r  =  having  red  blood,  etc. 

Given:  a  =  ir  and  acs  (n  +  t),  with  the  implied  condition,  n  t  =  0.     To 
eliminate  a  and  find  an  expression  for  s  t. 
25  See  Boole,  Laws  of  Thought,  pp.  144-46. 


208 


A  Survey  of  Symbolic  Logic 


a  =  iris  equivalent  to 

-(i  r)  a  +  -a  i  r  =  a  -i  +  a  -r  •§•  -a  i  r  =  0  (1) 

a  cs  (n  +  2)  is  equivalent  to  a--(s  n  +  s  t)  =0. 

-(*  n  +  s  f)  =  -(s  n)  --(s  i)  =  (-s  +  -n)(-s  +  -t)  =  -s  +  -n  -t. 

Hence,  a  -s  +  a  -n  -t  =  0  (2) 

Combining  (1)  and  (2)  and  n  t  =  0, 

a  ~i  +  a  -r  +  -a  i  r  +  a  -s  +  a  -n  ~t  +  n  t  =  0  (3) 

Eliminating  a,  by  7*4, 

(-i  +  -r  +  -s  +  -n  -t  +  n  t)  (i  r  +  n  t)  =  nt  +  ir  -s  +  ir  -n-t  =  0 

The  solution  of  this  equation  for  s  is26  ires. 
And  its  solution  for  t  is  i  r  -n  c  t  c  -n. 

Hence  [5-3]  ir  -nc.sic.-n,  t>r  st  =  ir-n  +  u--n,  where  u  is  un- 
determined. 
The  soft-bodied  animals  enclosed  in  a  tube  consist  of  the  invertebrates 


-i 


26  See  Chap,  n,  Sect,  iv,  "Symmetrical  and  Unsymmetrical  Constituents  of  an  Equa- 
tion ". 


Applications  of  the  Boole-Schroder  Algebra  209 

which  have  red  blood  in  a  double  system  of  circulating  vessels  and  a  body 
covering,  together  with  an  undetermined  additional  class  (which  may  be 
null)  of  other  animals  which  have  a  body  covering.  This  solution  may  be 
verified  by  the  diagram  of  equation  (3)  (figure  22).  In  this  diagram,  s  t  is 
the  square  formed  by  the  two  crossed  rectangles.  The  lower  half  of  this 
inner  square  exhibits  the  solution.  Note  that  the  qualification,  -n,  in 
ir  -nest,  is  necessary.  In  the  top  row  is  a  single  undeleted  area  repre- 
senting a  portion  of  i  r  (n)  which  is  not  contained  in  s  t. 

Example  7.27 

Demonstrate  that  from  the  premises  "All  a  is  either  b  or  c",  and 
"All  c  is  a",  no  conclusion  can  be  drawn  which  involves  only  two  of  the 
classes,  a,  b,  and  c. 

Given :  a  c  (b  +  c)  and  c  c  a. 

To  prove  that  the  elimination  of  any  one  element  gives  a  result  which 
is  either  indeterminate  or  contained  in  one  or  other  of  the  premises. 

a  c  (b  +  c)  is  equivalent  to  a-b  -c  =  0. 
And  c  c  a  is  equivalent  to  -a  c  =  0. 
Combining  these,  a  -b  -c  +  -a  c  =  0. 

Eliminating  a  [7-4],  (-b  -c)  c  =  0,  which  is  the  identity,  0  =  0. 
Eliminating  c,  (a  -b)  -a  =  0,  or  0  =  0. 

Eliminating   b,    (-a  c  +  a  -c)  -a  c  =  -ac  =  0,   which  is   the   second 
premise. 

Example  8. 

A  set  of  balls  are  all  of  them  spotted  with  one  or  more  of  the  colors,  red, 
green,  and  blue,  and  are  numbered.  And  all  the  balls  spotted  with  red  are 
also  spotted  with  blue.  All  the  odd-numbered  blue  balls,  and  all  the  even 
numbered  balls  which  are  not  both  red  and  green,  are  on  the  table.  De- 
scribe the  balls  not  on  the  table. 

Let  e  =  even-numbered,  -e  =  odd-numbered, 
r  =  spotted  with  red, 
b  =  spotted  with  blue, 
g  =  spotted  with  green, 
t  =  balls  on  the  table. 
Given:  (1)  -r-b-g  =  0. 

27  See  De  Morgan,  Formal  Logic,  p.  123. 
15 


210 


A  Survey  of  Symbolic  Logic 


(2)  r-6  =  0. 

(3)  [-eb  +  e  -(r  g)]  ct,    or    (-e  b  +  e  -r  +  e  -g)  -t  =  0. 

To  find  an  expression,  x,  such  that  -I  c  x,  or  -t  x  =  -t.  Such  an  expression 
should  be  as  brief  as  possible.  Consequently  we  must  develop  -t  with 
respect  to  e,  r,  b,  and  g,  and  eliminate  all  null  terms.  (An  alternative 
method  would  be  to  solve  for  -t,  but  the  procedure  suggested  is  briefer.) 

-t  =  -t(e  +  -e)  (r  +  -r}  (b  +  -b}  (g  +  -g) 

=  -t(erbg+erb-g+er-bg  +  e-rbg  +  -erbg  +  er-b-g 
+  e  -r  b  -g  +  -e  r  b  -g  +  e  -r  -b  g  +  -e  -r  b  g  +  -e  r  -b  g 
+  e-r  -b  -g  +  -e  r  -b  -g  +  -e-rb-g  +  -e  -r  -b  g  +  -e  -r  -b  -g)    (4) 
From  (1),  (2),  and  (3), 

-t  (-eb  +  e-r  +  e-g  +  r-b  +  -r  -b  -g)  =  0  (5) 

Eliminating  from  (4)  terms  involved  in  (5), 

-t  —  -t  (e  rb  g  +  -e-r  -b  g),    or    -tc(erbg  +  -e-r  -b  g) 

All  the  balls  not  on  the  table  are  even-numbered  and  spotted  with  all  three 
colors  or  odd-numbered  and  spotted  with  green  only. 


Applications  of  the  Boole-Schroder  Algebra  211 

In  the  diagram  (figure  23),  equation  (1)  is  indicated  by  vertical  lines, 
(2)  by  oblique,  (3)  by  horizontal. 

Example  9.28 

Suppose  that  an  analysis  of  the  properties  of  a  particular  class  of  sub- 
stances has  led  to  the  following  general  conclusions: 

1st.  That  wherever  the  properties  a  and  b  are  combined,  either  the 
property  c,  or  the  property  d,  is  present  also;  but  they  are  not  jointly  present. 

2d.  That  wherever  the  properties  b  and  c  are  combined,  the  properties 
a  and  d  are  either  both  present  with  them,  or  both  absent. 

3d.  That  wherever  the  properties  a  and  b  are  both  absent,  the  proper- 
ties -c  and  d  are  both  absent  also;  and  vice  versa,  where  the  properties 
c  and  d  are  both  absent,  a  and  b  are  both  absent  also. 

Let  it  then  be  required  from  the  above  to  determine  what  may  be  con- 
cluded in  any  particular  instance  from  the  presence  of  the  property  a  with 
respect  to  the  presence  or  absence  of  the  properties  b  and  c,  paying  no 
regard  to  the  property  d. 

Given:  (1)  a  b  c  (c  -d  +  -c  d). 

(2)  be  c  (ad  +  -a-d). 

(3)  -a  -b  =  -c  -d. 

To  eliminate  d  and  solve  for  a. 

(1)  is  equivalent  to  a  b--(c  -d  +  -c  d)  =  0. 

(2)  is  equivalent  to  b  c--(a  d  +  -a  -d)  =  0. 
But  [6  •  4]  -(c  -d  +  -c  d)  =  c  d  +  -c  -d, 

and  -(a  d  +  -a  -d)  =  -a  d+  a  -d. 

Hence  we  have,  a  b  (c  d  +  -c  -d}  =abcd+ab-c-d  =  0.  (4) 

and  b  c  (-a  d+  a  -d)  =  -abcd+abc-d  =  Q  (5) 

(3)  is  equivalent  to 

-a  -b  (c  +  d)  +  (a  +  6)  -c  -d 

=  -a  -b  c  +  -a  -6  d  +  a  -c  -d  +  b  -c  -d  =  0     (6) 

Combining  (4),  (5),  and  (6),  and  giving  the  result  the  form  of  a 
function  of  d, 

(-a  -b  c  +  -a  -b  +  a  b  c  +  -a  b  c)  d 

+  (-a  -b  c  +  a  -c  +  b  -c  +  a  b  -c  +  a  b  c)  -d  =  0 

28  See  Boole,  Laws  of  Thought,  pp.  118-20.  For  further  problems,  see  Mrs.  Ladd- 
Franklin,  loc.  tit.,  pp.  51-61,  Venn,  Symbolic  Logic,  Chap,  xm,  and  Schroder,  Algebra  der 
Logik:  Vol.  i,  Dreizehnte  Vorlesung. 


212 


A  Survey  of  Symbolic  Logic 


Or,  simplifying,  by  5-4  and  5-91, 

(-a  -b  +  b  c)  d  +  (-a  -b  c  +  a-c  +  b-c  +  ab  c)  -d  =  0 
Hence  [7  •  4]  eliminating  d, 

(-a  -l  +  5  c)  (-a  -b  c  +  a-c  +  b-c  +  ab  c)  =  -a-b  c  +  ab  c  =  0 
Solving  this  equation  for  a  [7-2],  -be  c  a  c  (-6  +  -c). 

The  property  a  is  always  present  when  c  is  present  and  b  absent,  and  when- 
ever a  is  present,  either  b  is  absent  or  c  is  absent. 

The  diagram  (figure  24)  combines  equations  (4),  (5),  and  (6). 


FIG.  24 

As  Boole  correctly  claimed,  the  most  powerful  application  of  this  algebra 
is  to  problems  of  probability.  But  for  this,  additional  laws  which  do  not 
belong  to  the  system  are,  of  course,  required.  Hence  we  omit  it.  Some- 
thing of  what  the  algebra  will  do  toward  the  solution  of  such  problems  will 
be  evident  if  the  reader  imagine  our  Example  8  as  giving  numerically  the 
proportion  of  balls  spotted  with  red,  with  blue,  and  with  green,  and  the 
quaesitum  to  be  "If  a  ball  not  on  the  table  be  chosen  at  random,  what  is 
the  probability  that  it  will  be  spotted  with  all  three  colors?  that  it  will  be 
spotted  with  green?"  The  algebra  alone,  without  any  additional  laws, 
answers  the  last  question.  As  the  reader  will  observe  from  the  solution, 
all  the  balls  not  on  the  table  are  spotted  with  green. 


Applications  of  the  Boole-Schroder  Algebra  213 

III.    THE  APPLICATION  TO  PROPOSITIONS 

If,  in  our  postulates,  a,  b,  c,  etc.,  represent  propositions,  and  the  "prod- 
uct", a  b,  represent  the  proposition  which  asserts  a  and  b  both,  then  we 
have  another  interpretation  of  the  algebra.  Since  a+b  is  the  negative  of 
-a-b,  a  +  b  will  represent  "It  is  false  that  a  and  b  are  both  false",  or 
"At  least  one  of  the  two,  a  and  b,  is  true".  It  has  been  customary  to  read 
a  +  b,  "Either  a  or  b",  or  "Either  a  is  true  or  6  is  true".  But  this  is  some- 
what misleading,  since  "Either  ...  or  ..."  frequently  denotes,  in 
ordinary  use,  a  relation  which  is  to  be  understood  in  intension,  while  this 
algebra  is  incapable  of  representing  relations  of  intension.  For  instance, 
we  should  hardly  affirm  "Either  parallels  meet  at  finite  intervals  or  all 
men  are  mortal".  We  might  well  say  that  the  "Either  .  .  .  or  .  .  ." 
relation  here  predicated  fails  to  hold  because  the  two  propositions  are 
irrelevant.  But  at  least  one  of  the  two,  "Parallels  meet  at  finite  intervals" 
and  "All  men  are  mortal",  is  a  true  proposition.  The  relation  denoted 
by  +  in  the  algebra  holds  between  them.  Hence,  if  we  render  a  +  b  by 
"Either  a  or  6",  we  must  bear  in  mind  that  no  necessary  connection  of  a 
and  b,  no  relation  of  "relevance"  or  "logical  import",  is  intended. 

The  negative  of  a,  -a,  will  be  its  contradictory,  or  the  proposition  "a  is 
false".  It  might  be  thought  that  -a  should  symbolize  the  "contrary" 
of  a  as  well, — that  if  a  be  "All  men  are  mortal ",  then  "No  men  are  mortal" 
should  be  -a.  But  if  the  contrary  as  well  as  the  contradictory  be  denoted 
by  -a,  then  -a  will  be  an  ambiguous  function  of  a,  whereas  the  algebra 
requires  that  -a  be  unique. 29 

The  interpretation  of  0  and  1  is  most  easily  made  clear  by  considering 
the  connection  between  the  interpretation  of  the  algebra  for  propositions 
and  its  interpretation  for  classes.  The  propositional  sign,  a,  may  equally 
well  be  taken  to  represent  the  class  of  cases  in  which  the  proposition  a  is 
true,  a  b  will  then  represent  the  class  of  cases  in  which  a  and  6  are  both 
true;  -a,  the  class  of  cases  in  which  a  is  false,  and  so  on.  The  "universe",  1, 
will  be  the  class  of  all  cases,  or  all  "actual"  cases,  or  the  universe  of  facts. 
Thus  a  =  1  represents  "The -cases  in  which  a  is  true  are  all  cases",  or 
"a  is  true  in  point  of  fact",  or  simply  "a  is  true".  Similarly  0  is  the  class 
of  no  cases,  and  a  =  0  will  mean  "a  is  true  in  no  case",  or  "a  is  false". 

It  might  well  be  asked:  May  not  a,  b,  c,  etc.,  represent  statements  which 
are  sometimes  true  and  sometimes  false,  such  as  "Today  is  Monday" 
or  "The  die  shows  an  ace"?  May  not  a  symbolize  the  cases  in  which  a  is 

29  See  Chap.  11,  3-3. 


214  A  Survey  of  Symbolic  Logic 

true,  and  these  be  not  all  but  only  some  of  the  cases?  And  should  not 
a  =  1  be  read  "a  is  always  true",  as  distinguished  from  the  less  com- 
prehensive statement,  "a  is  true"?  The  answer  is  that  the  interpretation 
thus  suggested  can  be  made  and  that  Boole  actually  made  it  in  his  chapters 
on  "Secondary  Propositions".30  But  symbolic  logicians  have  come  to 
distinguish  between  assertions  which  are  sometimes  true  and  sometimes 
false  and  propositions.  In  the  sense  in  which  "Today  is  Monday"  is 
sometimes  true  and  sometimes  false,  it  is  called  a  propositional  function 
and  not  a  proposition.  There  are  two  principal  objections  to  interpreting 
the  Boole-Schroder  Algebra  as  a  logic  of  propositional  functions.  In  the 
first  place,  the  logic  of  propositional  functions  is  much  more  complex  than 
this  algebra,  and  in  the  second  place,  it  is  much  more  useful  to  restrict  the 
algebra  to  propositions  by  the  additional  law  "If  a  =f=  0,  then  a  =  1,  and 
if  a  =f=  lj  then  a  =  0",  and  avoid  any  confusion  of  propositions  with  asser- 
tions which  are  sometimes  true  and  sometimes  false.  In  the  next  chapter, 
we  shall  investigate  the  consequences  of  this  law,  which  holds  for  proposi- 
tions but  not  for  classes  or  for  propositional  functions.  We  need  not  pre- 
sume this  law  at  present:  the  Boole-Schroder  Algebra,  exactly  as  presented 
in  the  last  chapter,  is  applicable  throughout  to  propositions.  But  wre  shall 
remember  that  a  proposition  is  either  always  true  or  never  true :  if  a  proposi- 
tion is  true  at  all,  it  is  always  true.  Hence  in  the  interpretation  of  the 
algebra  for  propositions,  a  =  1  means  "a  is  true"  or  "a  is  always  true" 
indifferently — the  two  are  synonymous.  And  a  =  0  means  either  "a  is 
false"  or  "a  is  always  false". 

The  relation  a  c  6,  since  it  is  equivalent  to  a  -b  =  0,  may  be  read  "  It 
is  false  that  'a  is  true  and  b  is  false'",  or  loosely,  "If  a  is  true,  then  b  is 
true ".  But  a  c  b,  like  a  +  b,  is  here  a  relation  which  does  not  signify 
"relevance"  or  a  connection  of  "logical  import".  Suppose  a  =  "2  +  2 
=  4"  and  b  =  "Christmas  is  a  holiday".  We  should  hardly  say  "If 
2  +  2  =  4,  then  Christmas  is  a  holiday".  Yet  it  is  false  that  "2  +  2  =  4 
and  Christmas  is  not  a  holiday":  in  this  example  a  -b  =  0  is  true,  and 
hence  acb  will  hold.  This  relation,  a  c  b,  is  called  "material  implication"; 
it  is  a  relation  of  extension,  whereas  we  most  frequently  interpret  "implies" 
as  a  relation  of  intension.  But  a  c  b  has  one  most  important  property  in 
common  with  our  usual  meaning  of  "a  implies  b" — when  a  cb  is  true,  the 
case  in  which  a  is  true  but  b  is  false  does  not  occur.  If  a  c  b  holds,  and  a  is 
true,  then  b  will  not  be  false,  though  it  may  be  irrelevant.  Thus  "material 

30  Laws  of  Thought,  Chaps,  xi-xiv. 


Applications  of  the  Boole-Schroder  Algebra  215 

implication"  is  a  relation  which  covers  more  than  the  "implies"  of  ordinary 
logic:  a  cb  holds  whenever  the  usual  "a  implies  b"  holds;  it  also  holds  in 
some  cases  in  which  "a  implies  b"  does  not  hold.31 

The  application  of  the  algebra  to  propositions  is  so  simple,  and  so 
resembles  its  application  to  classes,  that  a  comparatively  few  illustrations 
will  suffice.  We  give  some  from  the  elementary  logic  of  conditional  propo- 
sitions, and  conclude  with  one  taken  from  Boole. 

Example  1. 

If  A  is  B,  C  is  D.  (1) 

And  A  is  B.  (2) 

Let  x  =  A  is  B;  y  =  C  is  Z). 

The  two  premises  then  are : 

(1)  xcy,  or  [4-9]  -x  +  y  =  1. 

(2)  x  =  1,  or  -x  =  0. 

[5-7]  Since  -x  +  y  =  1  and  -x  =  0,  y  =  1. 
y  =  I  is  the  conclusion  "  C  is  D  ". 

Example  2. 

(1)  If  A  is  B,  C  is  Z). 

(2)  But  C  is  not  D. 

Let  x  =  A  is  B;  y  =  C  is  Z). 

(1)  x  cy,  or  -x  +  y  =  1. 

(2)  y  =  Q. 

[5-7]  Since  -x  +  y  =  1  and  y  =  0,  -x  =  1. 

-x  =  1  is  the  conclusion  "A  is  5  is  false",  or  "A  is  not  5". 

Example  3. 

(1)  If  ^  is  B,  C  is  Z);  and  (2)  if  EhF,G  is  H. 

(3)  But  either  ^4  is  B  or  C  is  D. 

Let  w  =  ^4  is  B;  x  =  C  is  D;  y  =  E  is  F;  z  =  G  is  H. 

(1)  wcx,  or  [4-9]  wx  =  w. 

(2)  i/cz,  or  yz  =  y. 

(3)  W  +  T/  =  1. 

31  "Material  implication"  is  discussed  more  at  length  in  Chap,  iv,  Sect,  i,  and  Chap, 
v,  Sect.  v. 


216 


A  Survey  of  Symbolic  Logic 


Since  iv  +  y  =  1,  and  wx  =  w  and  yz  =  y,wx  +  yz  =  l. 
Hence  [4-5]  w  x  +  -w  x  +  y  z  +  -y  z  —  1  +  -w  x  +  -y  z  =  1 . 
Hence  x  (w  +  -w)  +  z  (y  +  -y)  =  x  +  z  =  1. 

x+z  =  1  is  the  conclusion  "Either  C  is  D  or  G  is  H".  This  dilemma 
may  be  diagrammed  if  we  put  our  equations  in  the  equivalent  forms 
(1)  w  -x  =  0,  (2)  y  -z  =  0,  (3)  -w  -y  =  0.  In  figure  25,  w  -x  is  struck 


FIG.  25 

out  with  horizontal  lines,  y  -z  with  vertical,  -w  -y  with  oblique ,  That 
everything  which  remains  is  either  x  or  z  is  evident. 

Example  4. 

(1)  Either  A  is  B  or  C  is  not  D. 

(2)  Either  C  is  D  or  E  is  F. 

(3)  Either  A  is  B  or  E  is  not  F. 

Let  z  =  yl  is  £;  y  =  C  is  D;  z  =  E  is  F. 

(1)  ar  +  -y  =  1. 

(2)  y  +  z  =  1,  or  -y -z  =  0. 

(3)  z  +  -z  =  1,  or  -xz  =  0. 

By  (1),  x  +  -y(z  +  -z)  =  x  +  -y  z  +  -y  -z  =  1. 

Hence  by  (2),  x  +  -y  z  =  1  =  x  +  -y  z  (x  +  -x}  —  x  +  x  -y  z  +  -x  -y  z. 

And  by  (3),  -x  -y  z  =  0.     Hence  x  +  x-y  z  =  x  —  1. 

Thus  these  three  premises  give  the  categorical  conclusion  "A  is  B",  indi- 
cating the  fact  that  the  traditional  modes  of  conditional  syllogism  are  by 
no  means  exhaustive. 


Applications  of  the  Boole- Schroder  Algebra  217 

Example  5.32 

Assume  the  premises: 

1.  If  matter  is  a  necessary  being,  either  the  property  of  gravitation  is 
necessarily  present,  or  it  is  necessarily  absent. 

2.  If  gravitation  is  necessarily  absent,  and  the  world  is  not  subject  to 
any  presiding  intelligence,  motion  does  not  exist. 

3.  If  gravitation  is  necessarily  present,  a  vacuum  is  necessary. 

4.  If  a  vacuum  is  necessary,  matter  is  not  a  necessary  being. 

5.  If  matter  is  a  necessary  being,  the  world  is  not  subject  to  a  presiding 
intelligence. 

Let  x  =  Matter  is  a  necessary  being. 

y  =  Gravitation  is  necessarily  present. 
z  =  The  world  is  not  subject  to  a  presiding  intelligence. 
w  =  Motion  exists. 
t  =  Gravitation  is  necessarily  absent. 
v  =  A  vacuum  is  necessary. ' 

The  premises  then  are : 

(1)  x  c  (y  +  t},  or  x  -y  -t  =  0. 
^2)  t  z  c  -w,  or  t  z  w  =  0. 

(3)  y  cv,  or  y  -v  =  0. 

(4)  v  c  -x,  or  v  x  =  0. 

(5)  x  c  z,  or  x  -z  =  0. 

And  since  gravitation  cannot  be  both  present  and  absent, 

(6)  y  t  =  0. 
Combining  these  equations : 

x  -y  -t  +  t  z  iv  +  y  -v  +  v  x  +  x  -z  +  y  t  =  0  (7) 

From  these  premises,  let  it  be  required,  first,  to  discover  any  conection 
between  x,  "Matter  is  a  necessary  being",  and  y,  "Gravitation  is  necessarily 
present".  For  this  purpose,  it  is  sufficient  to  discover  whether  any  one 
of  the  four,  x  y  —  0,  x  -y  =  0,  -x  y  =  0,  or  -x  -y  =  0,  since  these  are 
the  relations  which  state  any  implication  which  holds  between  x,  or  -x, 
and  y,  or  -y.  This  can  always  be  done  by  collecting  the  coefficients  of 
x  y,  x  -y,  -x  y,  and  -x  -y,  in  the  comprehensive  expression  of  the  data, 
such  as  equation  (7),  and  finding  which  of  them,  if  any,  reduce  to  1.  But 

32  See  Boole,  Laws  of  Thought,  Chap.  xiv.  The  premises  assumed  are  supposed  to  be 
borrowed  from  Clarke's  metaphysics. 


218 


A  Survey  of  Symbolic  Logic 


sometimes,  as  in  the  present  case,  this  lengthy  procedure  is  not  necessary, 
because  the  inspection  of  the  equation  representing  the  data  readily  reveals 
such  a  relation. 

From. (7),  [5-72]  vx  +  -vy  =  0. 

Hence  [1-5]  V  x  y  +  -v  x  y  =  (v  +  -v)  xy  =  xy  =  Q,  orxc  -y,  y  c  -x. 
If  matter  is  a  necessary  being,  then  gravitation  is  not  necessarily  present; 
if  gravitation  is  necessarily  present,  matter  is  not  a  necessary  being. 

Next,  let  any  connection  between  x  and  w  be  required.  Here  no  such 
relation  is  easily  to  be  discovered  by  inspection.  Remembering  that  if 
a  =  0,  then  a  b  =  0  and  a  -b  =  0 ; 

From  (7),  (-y -t+tz  +  y-v  +  i)  +  -z  +  yt)wx 
+  (tz  +  y  -v  +  y  t)  w  -x 
+  (-y  -t  +  y  -v  +  v  +  -z  +  y  f)  -w  x 
+  (y-v  +  yf)  -w  -x  =  0  (8) 

Here  the  coefficient  of  w  x  reduces  to  1,  for  [5-85], 

y  -v  +  v  =  y  +  v,        and         t  z  +  -z  =  t  +  -z 

and  hence  the  coefficient  is  -y  -t  +  y  +  t  +  v  +  -z  +  y  t. 

But  [5-96]    (-y-t  +  y  +  t)  +  v  +  -z  +  yt  =  l+v  +  -z  +  yt  =  1. 

Hence  w  x  =  0,  or  w  c  -x,  x  c  -w. 


-IV 


--y-          --{ 

FIG.  26 


-v 


-t 


Applications  of  the  Boole-Schroder  Algebra  219 

None  of  the  other  coefficients  in  (8)  reduces  to  1.  Hence  the  conclusion 
which  connects  x  and  w  is:  "If  motion  exists,  matter  is  not  a  necessary 
being;  if  matter  is  a  necessary  being,  motion  does  not  exist". 

Further  conclusions,  relating  other  terms,  might  be  derived  from  the 
same  premises.  All  such  conclusions  are  readily  discoverable  in  the  dia- 
gram of  equation  (7).  In  fact,  the  diagram  is  more  convenient  for  such 
problems  than  the  transformation  of  equations  in  the  algebra. 

Another  method  for  discovering  the  implications  involved  in  given  data 
is  to  state  the  data  entirely  in  terms  of  the  relation  c ,  and,  remembering 
that  "If  acb  and  bcc,  then  ace",  as  well  as  "acb  is  equivalent  to 
-b  c-a",  to  seek  directly  any  connection  thus  revealed  between  the  propo- 
sitions which  are  in  question.  Although  by  this  method  it  is  possible  to 
overlook  a  connection  which  exists,  the  danger  is  relatively  small. 

IV.    THE  APPLICATION  TO  RELATIONS 

The  application  of  the  algebra  to  relations  is  relatively  unimportant, 
because  the  logic  of  relations  is  immensely  more  complex  than  the  Boole- 
Schroder  Algebra,  and  requires  more  extensive  treatment  in  order  to  be  of 
service.  We  shall,  consequently,  confine  our  discussion  simply  to  the 
explanation  of  this  interpretation  of  the  algebra. 

A  relation,  taken  in  extension,  is  the  class  of  all  couples,  triads,  or  tetrads, 
etc.,  which  have  the  property  of  being  so  related.  That  is,  the  relation 
"father  of"  is  the  class  of  all  those  couples,  (x;y],  such  that  x  is  father 
of  y:  the  dyadic  relation  R  is  the  class  of  all  couples  (x;  y)  such  that  x  has 
the  relation  R  to  y,  x  R  y.  The  extension  of  a  relation  is  the  class  of  things 
which  have  the  relation.  We  must  distinguish  between  the  class  of  couples 
(x;  y)  and  the  class  of  couples  (y;  x),  since  not  all  relations  are  symmetrical 
and  x  R  y  commonly  differs  from  y  R  x.  Since  the  properties  of  relations, 
so  far  as  the  laws  of  this  algebra  apply  to  them,  are  the  same  whether  they 
are  dyadic,  triadic,  or  tetradic,  etc.,  the  discussion  of  dyadic  relations  wrill  be 
sufficient. 

The  "product ",  R  x  S,  or  R  S,  will  represent  the  class  of  all  those  couples 
(x;  y}  such  that  x  R  y  and  x  S  y  are  both  true.  The  "sum ",  R  +  S,  will  be 
the  class  of  all  couples  (x ;  y}  such  that  at  least  one  of  the  two,  x  R  y  and 
x  S  y,  holds.  The  negative  of  R,  -R,  will  be  the  class  of  couples  (x;  y)  for 
which  x  R  y  is  false. 

The  null-relation,  0,  will  be  the  null-class  of  couples.  If  the  class  of 
couples  (t;  u)  for  which  t  R  u  is  true,  is  a  class  with  no  members,  and  the 


220  A  Survey  of  Symbolic  Logic 

class  of  couples  (v;  w)  for  which  v  S  w  is  true  is  also  a  class  with  no  members, 
then  R  and  S  have  the  same  extension.  It  is  this  extension  which  0  repre- 
sents. Thus  R  =  0  signifies  that  there  are  no  two  things,  i  and  u,  such 
that  i  R  u  is  true — that  nothing  has  the  relation  R  to  anything.  Similarly, 
the  universal-relation,  1,  is  the  class  of  all  couples  (in  the  universe  of  dis- 
course). 

The  inclusion,  RcS,  represents  the  assertion  that  every  couple  (x;  y) 
for  which  x  R  y  is  true  is  also  such  that  x  S  y  is  true;  or,  to  put  it  otherwise, 
that  the  class  of  couples  (x ;  y)  for  which  x  R  y  is  true  is  included  in  the 
class  of  couples  (u;  t>)  for  which  u  S  v  is  true.  Perhaps  the  most  satisfactory 
reading  of  R  cS  is  "The  presence  of  the  relation  R  implies  the  presence  of 
the  relation  S".  R  =  S,  being  equivalent  to  the  pair,  RcS  and  ScR, 
signifies  that  R  and  S  have  the  same  extension — that  the  class  of  couples 
(x;  y)  for  which  x  Ry  is  true  is  identically  the  class  of  couples  (u;  v)  for 
which  u  S  v  is  true. 

It  is  obvious  that  all  the  postulates,  and  hence  all  the  propositions,  of 
the  Boole-Schroder  Algebra  hold  for  relations,  so  interpreted. 

1-1  If  R  and  <S  are  relations  (that  is,  if  there  is  a  class  of  couples  (x;  y) 
such  that  x  Ry  is  true,  and  a  class  of  couples  (u;  v)  such  that  u  S  r.  is  true), 
then  R  xS  is  a  relation  (that  is,  there  is  a  class  of  couples  (w;  z)  such  that 
w  R  z  and  w  S  z  are  both  true) .  If  R  and  S  be  such  that  there  is  no  couple 
(w;  z)  for  which  w  R  z  and  w  S  z  both  hold,  then  R  x  S  is  the  null-relation,  0 
— i.  e.,  the  null-class  of  couples. 

1-2  The  class  of  couples  (x;  y)  for  which  x  R  y  and  x  R  y  both  hold  is 
simply  the  class  of  couples  for  which  x  R  y  holds. 

1  •  3  The  class  of  couples  denoted  by  R  x  S  is  the  same  as  that  denoted 
by  S  xR — namely,  the  class  of  couples  (x;  y)  such  that  x  Ry  and  x  S  y 
are  both  true. 

1-4  The  class  of  couples  (x;  y)  for  which  xRy,  x  S  y,  and  x  Ty  all 
hold  is  identically  the  same  in  whatever  order  the  relations  be  combined— 
i.  e.,  Rx(SxT)  =  (RxS)  xT, 

1-5  R  xO  =  0 — i.  e.,  the  product  of  the  class  of  couples  for  which  x  R  y 
holds  and  the  null-class  of  couples  is  the  null-class  of  couples. 

1-6  For  every  relation,  R,  there  is  a  relation  -R,  the  class  of  couples 
for  which  x  R  y  is  false,  and  -R  is  such  that : 

1-61  If  the  relation  Rx-S  is  null  (that  is,  if  there  is  no  couple  such 
that  x  Ry  is  true  and  x  S  y  is  false),  then  R  x  S  =  R  (that  is,  the  class  of 
couples  for  which  x  Ry  is  true  is  identically  the  class  of  couples  for  which 
x  Ry  and  x  S  y  are  both  true) ;  and 


Applications  of  the  Boole-Schroder  Algebra  221 

1-62  If  R  xS  =  R  and  R  x-S  =  R,  then  R  =  0— i.  e.,  if  the  class  of 
couples  for  which  x  R  y  and  x  S  y  are  both  true  is  identically  the  class  of 
couples  for  which  x  R  y  is  true,  and  if  also  the  class  of  couples  for  which 
x  R  y  is  true  and  x  S  y  is  false  is  identically  the  class  of  couples  for  which 
x  R  y  is  true,  then  the  class  of  couples  for  which  x  R  y  is  true  is  null. 

1-71=  -0 — i.  e.,  the  universal  class  of  couples  is  the  negative  of  the 
null-class  of  couples,  within  the  universe  of  discourse  of  couples. 

1-8  R  +  S  =  -(-Rx-S) — i.  e.,  the  class  of  couples  (x;y)  such  that 
at  least  one  of  the  two,  x  R  y  and  x  S  y,  is  true  is  the  negative  of  the  class 
of  couples  for  which  x  R  y  and  x  S  y  are  both  false. 

1-9  RxS  =  Ris  equivalent  to  R  c  S — i.  e.,  if  the  class  of  couples  (x;  y) 
for  which  x  R  y  and  x  S  y  are  both  true  is  identical  with  the  class  of  couples 
for  which  x  Ry  is  true,  then  the  presence  of  R  implies  the  presence  of  *S; 
and  if  the  presence  of  R  implies  the  presence  of  S,  then  the  class  of  couples 
(x ;  y}  for  which  x  R  y  is  true  is  identical  with  the  class  of  couples  for  which 
x  Ry  and  x  S  y  are  both  true.33 

33  For  a  further  discussion  of  the  logic  of  relations,  see  Chap,  iv,  Sect.  v. 


CHAPTER    IV 
SYSTEMS  BASED  ON  MATERIAL  IMPLICATION 

We  are  concerned,  in  the  present  chapter,  with  the  "calculus  of  propo- 
sitions" or  calculus  of  "material  implication",  and  with  its  extension  to 
prepositional  functions.  We  shall  discover  here  two  distinct  modes  of 
procedure,  and  it  is  part  of  our  purpose  to  set  these  two  methods  side  by  side. 

The  first  procedure  takes  the  Boole-Schroder  Algebra  as  its  foundation, 
interprets  the  elements  of  this  system  as  propositions,  and  adds  to  it  a 
postulate  which  holds  for  propositions  but  not  for  logical  classes.  The 
result  is  what  has  been  called  the  "Two-Valued  Algebra",  because  the 
additional  postulate  results  in  the  law:  For  any  x,  if  x  =f=  1,  then  x  =  0, 
and  if  x  =f=  0,  then  x  =  1.  This  Two- Valued  Algebra  is  one  form  of  the 
calculus  of  propositions.  The  extension  of  the  Two- Valued  Algebra  to 
propositions  of  the  form  <pxn,  where  xn  is  an  individual  member  of  a  class 
composed  of  Xi,  x?,  x3,  etc.,  gives  the  calculus  of  prepositional  functions. 
II  and  2  functions  have  a  special  significance  in  this  system,  and  the  relation 
of  "formal  implication",  Hx(<f>x  c\f/x~),  is  particularly  important.  In  terms 
of  it,  the  logical  properties  of  relations — including  the  properties  treated 
in  the  last  chapter  but  going  beyond  them — can  be  established.  This  is 
the  type  of  procedure  used  by  Peirce  and  Schroder. 

The  second  method — that  of  Principia  Mathematica — begins  with  the 
calculus  of  propositions,  or  calculus  of  material  implication,  in  a  form  which 
is  simpler  and  otherwise  superior  to  the  Two-Valued  Algebra,  then  pro- 
ceeds from  this  to  the  calculus  of  prepositional  functions  and  formal  impli- 
cation, and  upon  this  last  bases  not  only  the  treatment  of  relations  but  also 
the  "calculus  of  classes". 

It  is  especially  important  for  the  comprehension  of  the  whole  subject 
of  symbolic  logic  that  the  agreement  in  results  and  the  difference  of  method, 
of  these  two  procedures,  should  be  understood.  Too  often  they  appear  to 
the  student  simply  unrelated. 

I.    THE  TWO-VALUED  ALGEBRA1 

If  the  elements  a,  b,  ...  p,  q,  etc.,  represent  propositions,  and  a  x  b  or 
a  b  represent  the  joint  assertion  of  a  and  b,  then  the  assumptions  of  the 

1  See  Schroder,  Algebra  der  Logik:  n,  especially  Fiinfzehnte  Vorlesung.  An  excellent 
summary  is  contained  in  Schroder's  Abriss  (ed.  M tiller),  Teil  u. 

222 


Systems  Based  on  Material  Implication  223 

Boole-Schroder  Algebra  will  all  be  found  to  hold  for  propositions,  as  was 
explained  in  the  last  chapter.2  As  was  there  made  clear,  p  =  0  will  repre- 
sent "p  is  false",  and  p  =  1,  "p  is  true".  Since  0  and  1  are  unique,  it 
follows  that  any  two  propositions,  p  and  q,  such  that  p  =  0  and  q  =  0, 
or  such  that  p  =  1  and  q  =  1,  are  also  such  that  p  —  q.  p  =  q,  in  the 
algebra,  represents  a  relation  of  extension  or  "truth  value",  not  an  equiva- 
lence of  content  or  meaning. 

-p  symbolizes  the  contradictory  or  denial  of  p. 

The  meaning  of  p  +  q  is  readily  determined  from  its  definition, 

p  +  q  =  -(-p  -q) 

p  +  q  is  the  denial  of  "p  is  false  and  q  is  false",  or  it  is  the  proposition 
"At  least  one  of  the  two,  p  and  q,  is  true",  p  +  q  may  be  read  loosely, 
"Either  p  is  true  or  q  is  true".  The  possibility  that  both  p  and  q  should 
be  true  is  not  excluded. 

p  c  q  is  equivalent  to  p  q  =  p  and  to  p  -q  =  0.  p  c  q  is  the  relation  of 
material  implication.  We  shall  consider  its  properties  with  care  later  in 
the  section.  For  the  present,  we  may  note  simply  that  p  c  q  means  exactly 
"It  is  false  that  p  is  true  and  q  false".  It  may  be  read  "If  p  is  true,  q  is 
true",  or  " p  (materially)  implies  q". 

With  the  interpretations  here  given,  all  the  postulates  of  the  Boole- 
Schroder  Algebra  are  true  for  propositions.  Hence  all  the  theorems  will 
also  be  true  for  propositions.  But  there  is  an  additional  law  which  holds 
for  propositions: 

P  =  (P  =  1) 

"The  proposition,  p,  is  equivalent  to  'p  is  true'".  It  follows  immediately 
from  this  that 

-p  =  (-p  =  1)  =  (p  =  0) 

11 -p  is  equivalent  to  'p  is  false'".  It  also  follows  that  -p  =  -(p  =  1), 
and  hence 

-(p  =  1)  =  (p  =  0),         and        -(p  =  0)  =  (p  =  1) 

'"p  =  1  is  false '  is  equivalent  to  p  =  0",  and  '"p  =  0  is  false '  is  equivalent 
to  p  =  1".  Thus  the  calculus  of  propositions  is  a  two-valued  algebra: 
every  proposition  is  either  =  0  or  =  1,  either  true  or  false.  We  may,  then, 
proceed  as  follows:  All  the  propositions  of  the  Boole-Schroder  Algebra 

2  However,  many  of  the  theorems,  especially  those  concerning  functions,  eliminations, 
and  solutions,  are  of  little  or  no  importance  in  the  calculus  of  propositions. 


224  A  Survey  of  Symbolic  Logic 

which  were  given  in  Chapter  II  may  be  regarded  as  already  established  in 
the  Two-Valued  Algebra.  We  may,  then,  simply  add  another  division  of 
propositions — the  additional  postulate  of  the  Two-Valued  Algebra  and  the 
additional  theorems  which  result  from  it.  Since  the  last  division  of  the- 
orems in  Chapter  II  was  numbered  8-,  we  shall  number  the  theorems  of 
this  section  9  • . 

The  additional  postulate  is: 

9-01     For  every  proposition  p,  p  =  (p  =  1). 

And  for  convenience  we  add  the  convention  of  notation : 
9-02     -(p  =  q)  is  equivalent  to  p  4=  <?• 

As  a  consequence  of  9-01,  we  shall  have  such  expressions  as  -(p  =  1)  and 
-(p  =  0).  9-02  enables  us  to  use  the  more  familiar  notation,  p  4=  1  and 
p*0. 

It  follows  immediately  from  9-01  that  the  Two-Valued  Algebra  cannot 
be  viewed  as  a  wholly  abstract  mathematical  system.  For  whatever  p 
and  1  may  be,  p  =  I  is  a  proposition.  Hence  the  postulate  asserts  that 
any  element,  p,  in  the  system,  is  a  proposition.  But  even  a  necessary 
interpretation  may  be  abstracted  from  in  one  important  sense — no  step  in 
proof  need  be  allowed  to  depend  upon  this  interpretation.  This  is  the 
procedure  we  shall  follow,  though  it  is  not  the  usual  one.  It  will  appear 
shortly  that  the  validity  of  the  interpretations  can  be  demonstrated  within 
the  system  itself. 

In  presenting  the  consequences  of  9-01  and  9-02,  we  shall  indicate 
previous  propositions  by  which  any  step  in  proof  is  taken,  by  giving  the 
number  of  the  proposition  in  square  brackets.  Theorems  of  Chapter  II 
may,  of  course,  be  used  exactly  as  if  they  were  repeated  in  this  chapter. 

9-1     -p=  (p  =  0). 

[9-01]  -p  =  (-p  =  1).     And  [3-2]  -p  =  1  is  equivalent  to  p  =  0. 
9-12     -p='(p4:l). 

[9-01]  p  =  (p  =  1).     Hence  [3-2]  -p  =  -(p  =  1)  =  (p  =j=  1). 
9-13     (p  *  1)  =  (p  =  0). 

[9-1-12] 

9-14     (p*0)  =  (p=  1). 
[9-13,3-2] 

9  •  13  and  9  •  14  together  express  the  fact  that  the  algebra  is  two-valued. 
Every  proposition  is  either  true  or  false. 


Systems  Based  on  Material  Implication  225 

Up  to  this  point — that  is,  throughout  Chapter  II — we  have  written  the 
logical  relations  "If  .  .  .  ,  then  .  .  .",  " Either  ...  or -...",  "Both 
.  .  .  and  .  .  .",  etc.,  not  in  the  symbols  of  the  system  but  just  as  they 
would  be  written  in  arithmetic  or  geometry  or  any  other  mathematical 
system.  We  have  had  no  right  to  do  otherwise.  That  "...  c  ..." 
is  by  interpretation  "If  .  .  .  ,  then  .  .  .",  and  ".  .  .  +  .  .  ."  is  by  inter- 
pretation "Either  .  .  .  or  .  .  .",  does  not  warrant  us  in  identifying  the 
theorem  "If  acb,  then  -bc-a"  with  "  (a  c6)  c  (•<-&  c-a) ".  We  have 
had  no  more  reason  to  identify  "If  .  .  .  ,  then  .  .  ."in  theorems  with 
"...  c  ..."  than  a  geometrician  would  have  to  identify  the  period  at 
the  end  of  a  theorem  with  a  geometrical  point.  The  framework  of  logical 
relations  in  terms  of  which  theorems  are  stated  must  be  distinguished  from 
the  content  of  the  system,  even  when  that  content  is  logic. 

But  we  can  now  prove  that  we  have  a  right  to  interchange  the  joint 
assertion  of  p  and  q  with  p  xg,  "If  p,  then  q",  with  pcq,  etc.  We  can 
demonstrate  that  if  p  and  q  are  members  of  the  class  K,  then  p  c  q  is  a 
member  of  K,  and  that  "If  p,  then  q",  is  equivalent  to  p  cq.  And  we  can 
demonstrate  that  this  is  true  not  merely  as  a  matter  of  interpretation  but 
by  the  necessary  laws  of  the  system  itself.  We  can  thus  prove  that  writing 
the  logical  relations  involved  in  the  theorems — "Either  .  .  .  or  .  .  .," 
"Both  .  .  .  and  .  .  .,"  "If  .  .  .  ,  then  .  .  ." — in  terms  of  +,  x,  c, 
etc.,  is  a  valid  procedure. 

The  theorems  in  which  these  things  are  proved  are  never  needed  here- 
after, except  in  the  sense  of  validating  this  interchange  of  symbols  and  their 
interpretation.  Consequently  we  need  not  give  them  any  section  number. 

(1)  If  p  is  an  element  in  K,  p  =  I  and  p  =  0  are  elements  in  K. 

[9-01]  If  p  is  an  element  in  K,  p  =  1  is  an  element  in  K.  [1-6] 
If  p  is  an  element  in  K,  -p  is  an  element  in  K,  and  hence  [9  •  1]  p  =  0 
is  an  element  in  A'. 

(2)  The  two,  p  and  q,  are  together  equivalent  to  p  x  q,  or  p  q. 

[9-01]  p  q  =  (p  q  =  1).  [5-73]  pq  =  1  is  equivalent  to  the 
pair,  p  =  1  and  q  =  1,  and  hence  [9-01]  to  the  pair,  p  and  q. 

(3)  If  p  and  q  are  elements  in  K,  then  p  c  q  is  an  element  in  K. 

[4-9]  p  cq  is  equivalent  to  p  -q  =  0,  and  hence  [9-1]  to  -(p-q), 
But  if  p  and  q  are  elements  in  K,  [1  •  6,  1  •  11  -(p  -q)  is  an  element  in  A'. 

(4)  -p  is  equivalent  to  "p  is  false". 

[9-12]  -p  =  (p  4=  1),  and  [8-01]  p  4=  1  is  equivalent  to  "p  =  I 
is  false",  and  hence  [9-01]  to  "p  is  false". 
16 


226  A  Survey  of  Symbolic  Logic 

(5)  peg  is  equivalent  to  "If  p,  then  g". 

[5-64]  pcq  gives   "If  p  =  1,  then  q  =  1",  and  hence  [9-01] 
"If  p,  then  g". 

And  "If  p,  then  g"  gives  peg,  for  [9-01]  it  gives  "If  p  =  1,  then 
g  =  1 ",  and 

(a)  Suppose  as  a  fact  p  =  1.     Then,  by  hypothesis,  g  =  1,  and 
[2-2]  pcq. 

(b)  Suppose  that  p  4=  1.     Then  [9-14]  p  =  0,  and  [5-63]  peg. 

(6)  If  p  and  g  are  elements  in  X,  then  p  =  q  is  an  element  in  K. 

[7-1]  p  =  g  is  equivalent  to  p-q  +  -pq  =  0,  and  hence  [9-1] 
to-(p-g  +  -pg).     Hence  [1-6,  1-1,  3-35]  Q.E.D. 

(7)  p  =  g  is  equivalent  to  "p  is  equivalent  to  g". 

[2  •  2]  p  =  g  is  equivalent  to  "peg  and  g  c  p  ". 

By  (5)  above,  "peg  and  g  cp"  is  equivalent  to  "If  p,  then  g,  and 
if  g,  then  p".  And  this  is  equivalent  to  "p  is  equivalent  to  g". 

(8)  If  p  and  g  are  elements  in  K,  then  p  =J=  5"  is  an  element  in  X. 

[9-02]  (p  +  g)  =  -(p  =  g). 
Hence,  by  (6)  above  and  1-6,  Q.E.D. 

(9)  p  4=  g  is  equivalent  to  "p  is  not  equivalent  to  g". 

By  (4)  and  (2)  above,  Q.E.D. 

(10)  p  +  g  is  equivalent  to  "At  least  one  of  the  two,  p  and  g,  is  true. 

[l-8]p  +  g  =  -(-p-g). 

By  (4)  and  (2)  above,  -(-p-g)  is  equivalent  to  "It  is  false  that 
(p  is  false  and  g  is  false) ".  And  this  is  equivalent  to  "At  least  one 
of  the  two,  p  and  g,  is  true". 

In  consideration  of  the  above  theorems,  we  can  henceforth  write  "... 
c  ..."  for  "If  .  .  .  ,  then  ...","...  =  ..."  for  ".  .  .  is  equivalent 
to  ...","...  +  ..."  for  "Either  .  .  .  or  .  .  .",  etc.,  for  we  have 
proved  that  not  only  all  expressions  formed  from  elements  in  K  and  the 
relations  x  and  +  are  elements  in  K,  but  also  that  expressions  which  in- 
volve c,  and  =,  and  4=  are  elements  in  the  system  of  the  Two-Valued 
Algebra.  The  equivalence  of  "If  .  .  .  ,  then  ..."  with  ".  .  .  c  .  .  .", 
of  "Both  .  .  .  and  ..."  with  ".  .  .  x  .  .  .",  etc.,  is  no  longer  a  matter 
of  interpretation  but  a  consequence  of  9-01,  p  =  (p  =  1).  Also,  we  can 
go  back  over  the  theorems  of  Chapter  II  and,  considering  them  as  propositions 
of  the  Two-Valued  Algebra,  we  can  replace  "If  .  .  .  ,  then  .  .  .",  etc., 


Systems  Based  on  Material  Implication  227 

by  the  symbolic  equivalents.  Each  theorem  not  wholly  in  symbols  gives  a 
corresponding  theorem  which  is  wholly  in  symbols.  But  when  we  consider 
the  Boole-Schroder  Algebra,  without  the  additional  postulate,  9-01,  this 
procedure  is  not  valid.  It  is  valid  only  where  9-01  is  one  of  the  postulates — 
i.  e.,  only  in  the  system  of  the  Two-Valued  Algebra. 

Henceforth  we  shall  write  all  our  theorems  with  p  c  q  for  "If  p,  then  q", 
p  =  q  for  "p  is  equivalent  to  q",  etc.  But  in  the  proofs  we  shall  frequently 
use  "If  .  .'.  ,  then  ..."  instead  of  ".  .  .  c  .  .  .  ",  etc.,  because  the 
symbolism  sometimes  renders  the  proof  obscure  and  makes  hard  reading. 
(That  this  is  the  case  is  due  to  the  fact  that  the  Two-Valued  Algebra  does 
not  have  what  we  shall  hereafter  explain  as  the  true  "logistic"  form.) 

9-15     0  4=  1. 

0=0.     Hence  [9- 13]  0  +  1. 

9-16     (p  +  ?)  =  (-P  =  q)  =  (p  =  -q). 

(1)  If  p  =  q  and  p  =  1,  then  q  +  1  and  [9-13]  q  =  0. 
And  if  p  =  1,  [3-2]  -p  =  0.     Hence  -p  =  q. 

(2)  If  p  4=  q  and  p  +  1,  then  [9-13]  p  =  0,  and  [3-2]  -p  =  1. 
Hence  if  p  =(=  q,  then  q  =t=  0,  and  [9  •  14]  q  =  1  =  -p. 

(3)  If  -p  =  q  and  q  =  1,  then  -p  =  1,  and  [3-2]  p  =  0. 
Hence  [9 -15]  2/4=  q. 

(4)  If  -p  =  q  and  q  =j=  1,  then  -p  4=  1,  and  [9-13]  -p  =  0. 
Hence  [3-2]  p  =  1,  and  p  4=  q. 

By  (1)  and  (2),  it  p  *  q,  then  -p  =  q.     And  by  (3)  and  (4),  if 
-p  =  q,  then  p  4=  g.     Hence  p  =J=  q  and  -p  =  q  are  equivalent. 
And[3-2](-p  =  g)  =  (p  =  -q). 

This  theorem  illustrates  the  meaning  of  the  relation,  =,  in  the  calculus 
of  material  implication.  If  p  4=  q,  then  either  p  =  1  and  q  =  0  or  p  =  0 
and  q  =  1.  But  if  p  =  1,  then  -p  =  0,  and  if  p  =  0,  then  -p  =  1.  Hence 
the  theorem.  Let  p  represent  "Caesar  died",  and  q  represent  "There  is 
no  place  like  home".  If  "Caesar  died"  is  not  equivalent  to  "There  is 
no  place  like  home",  then  "Caesar  did  not  die"  is  equivalent  to  "There 
is  no  place  like  home".  The  equivalence  is  one  of  truth  values — {  =  0}  or 
{  =  1} — not  of  content  or  logical  significance. 

9-17     p  =  (p  =  1)  =  (p  *  0)  =  (-p  =  0)  =  (-p  4=  1). 

[9-OM3-14-16] 
9-18     -p  =  (p  =  0)  =  (p  4=  1)  =  (-p  =  1)  =  (-p  4=  0). 

[9-M3-14-16] 


228  A  Survey  of  Symbolic  Logic 

9-2     (p  =  l)(p  =  0)  =  0. 

[2-4]  p-p  =  0.     And  [9-01]  p  =  (p  =  1);    [9-1]  -p  =  (p  =  0). 
No  proposition  is  both  true  and  false. 

9-21     (p*  I)(p4=0)  =0. 

[2-4]  -pp  =  0.     And  [9-18]  -p  =  (p  4=  1);   [9-17]  p  =  (p  =t=  0). 
9-22     (p  =  l)  +  (p  =  0)  =  1. 

[4-8]p  +  -p  =  l.     Hence  [9-01-1]  Q.  E.  D. 
Every  proposition  is  either  true  or  false. 

9-23     (p*  l)  +  (p*0)  =  1. 

[4-8,9-01-1] 

Theorems  of  the  same  sort  as  the  above,  the  proofs  of  which  are  obvious, 
are  the  following: 

9-24     (pg)  =  (pg=l)  =  (pg4:0)  =  (p  =  l)(g  -  1)  -  (p  +  0)(g  4=  0) 
=  (p  *  0)(g  -  1)  -  (p  =  l)(g  4=  0)  =  -(-p  +  -g) 
=  (-p  +  -g  =  0)  =  [(p  =  0)  +  (g  =  0)  =0] 
=  [(p  *  1)  +  (q  4=  1)  =  0],  etc.,  etc. 

9-25     (p  +  g)  _(p  +  9-l)  -  (p+g  =1=0)  =  (p  =  1)  +  (9  =  1) 

=  (P  *  0)  +  (g  *  0)  =  -(-p-g)  =  [(p  =  0)(g  =  0)  =  0] 
=  [(P  *  l)(g  =*=  1)  4=  1],  etc.,  etc. 

These  theorems  illustrate  the  variety  of  ways  in  which  the  same  logical 
relation  can  be  expressed  in  the  Two-Valued  Algebra.  This  is  one  of  the 
defects  of  the  system — its  redundancy  of  forms.  In  this  respect,  the 
alternative  method,  to  be  discussed  later,  gives  a  much  neater  calculus  of 
propositions. 

We  turn  now  to  the  properties  of  the  relation  c  .  We  shall  include  here 
some  theorems  which  do  not  require  the  additional  postulate,  9-01,  for  the 
sake  of  bringing  together  the  propositions  which  illustrate  the  meaning  of 
"material  implication". 

9-3      (peg)  =  (-p+g)  =  (p-g  =  0). 

[4-9]  (peg)  =  (p-g  =  0)  =  (-p  +  g  =  1). 
[9-01]  (-p  +  g  =  1)  =  (-p  +  g). 

"p  materially  implies  g"  is  equivalent  to  "Either  p  is  false  or  g  is  true", 
and  to  "It  is  false  £hat  p  is  true  and  g  false". 

Since  peg  has  been  proved  to  be  an  element  in  the  system,  "  It  is  false 
that  p  materially  implies  g"  may  be  symbolized  by  -(p  c  g). 


Systems  Based  on  Material  Implication  229 

9-31      -(peg)  =  (-p  +  q  =  0)  =  (p-g). 

[3-4]  -(-p+g)  =  p-q.     And  [9-3]  -(peg)  =  -(-p+g). 
[9.02]  -(-p  +  g)  =  (-p+q  =  0). 

"p  does  not  materially  imply  q"  is  equivalent  to  "It  is  false  that  either  p 
is  false  or  g  is  true",  and  to  "p  is  true  and  g  false". 

9-32     (p  =  0)  c(pcg). 

[5-63]  Ocg.     Hence  Q.E.D. 

If  p  is  false,  then  for  any  proposition  g,  p  materially  implies  g.  This  is 
the  famous — or  notorious — theorem:  "A  false  proposition  implies  any 
proposition". 

9-33     (g=l)c(pcg). 

[5-61]  pel.     Hence  Q.E.D. 

This  is  the  companion  theorem:  "A  true  proposition  is  implied  by  any 
proposition". 

9-34     -(p  eg)  c  (p  =  1). 

The  theorem  follows  from  9-32  by  the  reductio  ad  absurdum, 
since  if  -(peg),  then  [9-32]  p  4=  0,  and  [9-14]  p  =  1. 

If  there  is  any  proposition,  g,  which  p  does  not  materially  imply,  then  p  is 
true.  This  is  simply  the  inverse  of  9-32.  A  similar  consequence  of  9  •  33  is : 

9-35     -(peg)  c(g  =  0). 

If  -(peg),  then  [9-33]  g  4=  1,  and  [9-13]  g  =  0. 
If  p  does  not  materially  imply  g,  then  g  is  false. 

9-36     -(p  c  g)  c  (p  c  -g) ;    -(p  c  g)  c  (-p  c  g) ;    -(p  c  g)  c  (-p  c  -g). 

[9-34-35]  If  -(peg),  then  p  =  1  and  g  =  0. 
[3-2]  If  p  =  1,  -p  =  0,  and  if  g  =  0,  then  -g  =  1. 
[9  •  32]  If  -p  =  0,  then  -peg  and  -p  c  -g. 
[9-331  If  -g  =  1,  then  pc-g. 

If  p  does  not  materially  imply  g,  then  p  materially  implies  the  negative, 
or  denial,  ef  g,  and  the  negative  of  p  implies  g,  and  the  negative  of  p  implies 
the  negative  of  g.  If  "Today  is  Monday"  does  not  materially  imply 
"The  moon  is  made  of  green  cheese",  then  "Today  is  Monday"  implies 
"The  moon  is  not  made  of  green  cheese",  and  "Today  is  not  Monday" 
implies  "The  moon  is  made  of  green  cheese",  and  "Today  is  not  Monday" 
implies  "The  moon  is  not  made  of  green  cheese". 

Some  of  the  peculiar  properties  of  material  implication  are  due  to  the 


230  A  Survey  of  Symbolic  Logic 

fact  that  the  relations  of  the  algebra  were  originally  devised  to  represent 
the  system  of  logical  classes.  But  9-36  exhibits  properties  of  material 
implication  which  have  no  analogy  amongst  the  relations  of  classes.  9-36 
is  a  consequence  of  the  additional  postulate,  p  =  (p  =  1).  For  classes,  c 
represents  "is  contained  in":  but  if  a  is  not  contained  in  b,  it  does  not 
follow  that  a  is  contained  in  not-6 — a  may  be  partly  in  and  partly  outside 
of  6. 

9-37  -(pcq)c(qcp). 

[9  •  36]  If  -(p  c  q),  then  -p  c  -q,  and  hence  [3-1]  q  c  p. 

Of  any  two  propositions,  p  and  q,  if  p  does  not  materially  imply  q,  then  q 
materially  implies  p. 

9-38     (pq)c[(pcq)(qcp)]. 

[9-24]  pq  =  (p  =  l)(g  =  1).     Hence  [9-33]  Q.E.D. 
If  p  and  q  are  both  true,  then  each  materially  implies  the  other. 

9-39     (-p-q)c[(pcq)(qcp-)}. 

[9-24]  -p-q  =  (-p  =  l)(-g  =  1)  =  (p  =  0)fo  =  0). 
Hence  [9-32]  Q.E.D. 

If  p  and  q  are  both  false,  then  each  materially  implies  the  other. 
For  any  pair  of  propositions,  p  and  q,  there  are  four  possibilities: 

1)  p  =  1,  q  =  1:  p  true,  q  true. 

2)  P  =  0,  q  =  0:  p  false,  q  false. 

3)  p  =  0,  q  =  1 :  p  false,  q  true. 

4)  p  =  I,  q  =  0:  p  true,  q  false. 

Now  in  the  algebra,  0  cO,  1  c  1,  and  0  c  1;  but  1  cO  is  false.  Hence  in 
the  four  cases,  above,  the  material  implications  and  equivalences  are  as 
follows : 

1)  pcq,  qcp,  p  =  q. 

2)  pcq,  qcp,  p  =  q. 

3)  pcq,  -(qcp),  p  4=  q. 

4)  -(pcq),  qcp,  p  +  q. 

This  summarizes  theorems  9-31-9-39.  These  relations  hold  regardless  of 
the  content  or  meaning  of  p  and  q.  Thus  pcq  and  p  =  q  are  not  the 
"implication"  and  "equivalence"  of  ordinary  logic,  because,  strictly  speak- 
ing, p  and  q  in  the  algebra  are  not  "propositions"  but  simply  the  "truth 
values"  of  the  propositions  represented.  In  other  words,  material  impli- 


Systems  Based  on  Material  Implication  231 

cation  and  material  equivalence  are  relations  of  the  extension  of  proposi- 
tions, whereas  the  "implication"  and  "equivalence"  of  ordinary  logic  are 
relations  of  intension  or  meaning.  But,  as  has  been  mentioned,  the  material 
implication,  p  c  q,  has  one  most  important  property  in  common  with  "  q 
can  be  inferred  from  p"  in  ordinary  logic;  if  p  is  true  and  q  false,  pcq 
does  not  hold.  And  the  relation  of  material  equivalence,  p  =  q,  never 
connects  a  true  proposition  with  a  false  one. 

These  theorems  should  make  as  clear  as  it  can  be  made  the  exact 
meaning  and  character  of  material  implication.  This  is  important,  since 
many  theorems  whose  significance  would  otherwise  be  very  puzzling  follow 
from  the  unusual  character  of  this  relation. 

Two  more  propositions,  of  some  importance,  may  be  given : 

9-4     (pqcr)  =  (qpcr)  =  [pc(qcr)]  =  [qc(pcr)]. 

[1-3]   pq  =  qp-     Hence   [3-2]   -(p  q)  =  -(qp),   and   [-(p  q)  +  r] 

=  Hq  p)  +  r]. 

But  [9  •  3]  [-(p  q)  +  r]  =  (p  q  c  r),  and  [-(q  p)  +  r]  =  (qpcr). 

And  [3-41]  [-(p  q)  +  r]  =  [(-p  +  -q)  +  r]  =  [-p  +  (-q  +  r)]  =  [p  c  (q  c  r)] 

Similarly,  [-(q  p)  +  r]  =  [qc(pcr)]. 

This  theorem  contains  Peano's  Principle  of  Exportation, 

[(p  q)cr]c[pc(qc r)] 

"If  pq  implies  r,  then  p  implies  that  q  implies  r";  and  his  Principle  of 
Importation, 

[P  <=  (q  c  r)]  c  [(p  q}  c  r] 
"  If  p  implies  that  q  implies  r,  then  if  p  and  q  are  both  true,  r  is  true. " 

9-5     [(p  q)  c  r]  =  [(p  -r}  c  -q]  =  [(q  -r)  c  -p]. 

[9  •  3]  [(p  q)cr]  =  [-(p  q)  +  r]  =  [(-p  +  -g)  +  r]  =  [(-p  +  r)  +  -q] 

=  [(~q  +  r)  +  -p]  =  [~(p  -r)  +  -q]  =  [-(q  -r)  +  -p]. 
[9-3]  [-(p-r)+-q]  =  [(p-r)  c-q],  and 

[~(q-r)  +  -p]  =  [(q-r)  c-p]. 

If  p  and  q  together  imply  r,  then  if  p  is  true  but  r  is  false,  q  must  be  false, 
and  if  q  is  true  but  r  is  false,  p  must  be  false.  This  is  a  principle  first  stated 
by  Aristotle,  but  especially  important  in  Mrs.  Ladd-Franklin's  theory  of 
the  syllogism. 

We  have  now  given  a  sufficient  number  of  theorems  to  characterize  the 
Two-Valued  Algebra — to  illustrate  the  consequences  of  the  additional 


232  A  Survey  of  Symbolic  Logic 

postulate  p  =  (p  =  1),  and  the  properties  of  peg.     Any  further  theorems 
of  the  system  will  be  found  to  follow  readily  from  the  foregoing. 

A  convention  of  notation  which  we  shall  make  use  of  hereafter  is  the 
following:  A  sign  =,  unless  enclosed  in  parentheses,  takes  precedence  over 
any  other  sign;  a  sign  c,  unless  enclosed  in  parentheses,  takes  precedence 
over  any  +  or  x  ;  and  the  sign  + ,  unless  enclosed  in  parentheses,  takes 
precedence  over  a  relation  x .  This  saves  many  parentheses  and  brackets. 

II.    THE  CALCULUS  OF  PROPOSITIONAL  FUNCTIONS.    FUNCTIONS  OF  ONE 

VARIABLE 

The  calculus  of  prepositional  functions  is  an  extension  of  the  Two- 
Valued  Algebra  to  propositions  which  involve  the  values  of  variables.  Fol- 
lowing Mr.  Russell,3  we  may  distinguish  propositions  from  prepositional 
functions  as  follows:  A  proposition  is  any  expression  which  is  either  true 
or  false;  a  prepositional  function  is  an  expression,  containing  one  or  more 
variables,  which  becomes  a  proposition  when  each  of  the  variables  is  re- 
placed by  some  one  of  its  values. 

There  is  one  meaning  of  "Today  is  Monday"  for  which  'today'  denotes 
ambiguously  Jan.  1,  or  Jan.  2,  or  .  .  .  ,  etc.  For  example,  when  we  say 
"'Today  is  Monday'  implies  'Tomorrow  is  Tuesday'",  we  mean  that  if 
Jan.  1  is  Monday,  then  Jan.  2  is  Tuesday;  if  Jan.  2  is  Monday,  then  Jan. 
3  is  Tuesday;  if  July  4  is  Monday,  then  July  5  is  Tuesday,  etc.  'Today' 
and  'tomorrow'  are  here  variables,  whose  values  are  Jan.  1,  Jan.  2,  Jan.  3, 
etc.,  that  is,  all  the  different  actual  days.  When  'today'  is  used  in  this 
variable  sense,  "Today  is  Monday"  is  sometimes  true  and  sometimes  false, 
or  more  accurately,  it  is  true  for  some  values  of  the  variable  'today ',  and 
false  for  other  values.  "Today  is  Monday"  is  here  a  propositional  function. 

There  is  a  quite  different  meaning  of  "Today  is  Monday"  for  which 
'today'  is  not  a  variable  but  denotes  just  one  thing — Jan.  22,  1916.  In 
this  sense,  if  "Today  is  Monday"  is  true  it  is  always  true.  It  is  either 
simply  true  or  simply  false:  its  meaning  and  its  truth  or  falsity  cannot 
change.  For  this  meaning  of  'today ',  " Today  is  Monday  "  is  a  proposition. 
'Today, '  meaning  Jan.  16,  1916,  is  one  value  of  the  variable  'today'.  When 
this  value  is  substituted  for  the  variable,  then  the  propositional  function  is 
turned  into  a  proposition. 

3  See  Principles  of  Mathematics,  Chap,  vu,  and  Principia  Mathematica,  i,  p.  15.  Mr. 
Russell  carries  out  this  distinction  in  ways  which  we  do  not  follow.  But  so  far  as  is  here 
in  question,  his  view  is  the  one  we  adopt.  Principia  Mathematica  is  cited  hereafter  as 
Principia. 


Systems  Based  on  Material  Implication  233 

We  may  use  <px,  $(x,  y},  $(x,  y,  z),  etc.,  to  represent  prepositional 
functions,  in  which  the  variable  terms  are  x,  or  x  and  y,  or  x,  y,  and  z,  etc. 
These  propositional  functions  must  be  carefully  distinguished  from  the 
functions  discussed  in  Chapter  II.  We  there  used  /,  F,  and  the  Greek 
capitals,  $,  ^,  etc.,  to  indicate  functions;  here  we  use  only  Greek  small 
letters.  Also,  for  any  function  of  one  variable,  we  here  omit  any  parenthesis 
around  the  variable — <px,  \[/y,  £x. 

f(x],  ^(x,  y},  etc.,  in  Chapter  II  are  confined  to  representing  such 
expressions  as  can  be  formed  from  elements  in  the  class  K  and  the  relations 
x  and  +  .  If  £  and  y  in  ty(x,  y}  are  logical  classes,  then  ^(x,  y)  is  some 
logical  class,  such  as  x  +  y  or  a  x  +  b  -y.  Or  if  x  in  /(#)  is  a  proposition, 
then  f(x)  is  some  proposition  such  as  a  x  or  -x  +  b.  The  propositional 
functions,  <px,  \f/(x,  ?/),  f  (x,  y,  z),  etc.,  are  subject  to  no  such  restriction. 
<px  becomes  a  proposition  when  x  is  replaced  by  one  of  its  values,  but  it 
does  not  necessarily  become  any  such  proposition  as  ax  or  -x  +  b.  'x  is 
Monday, '  '  x  is  a  citizen  of  y, '  '  y  is  between  x  and  z ' — these  are  typical 
propositional  functions.  They  are  neither  true  nor  false,  but  they  become 
either  true  or  false  as  soon  as  terms  denoting  individual  things  are  sub- 
stituted for  the  variables  x,  y,  etc.  All  the  functions  in  this  chapter  are 
such  propositional  functions,  or  expressions  derived  from  them. 

A  fundamental  conception  of  the  theory  of  propositional  functions  is 
that  of  the  "range  of  significance".  The  range  of  significance  of  a  function 
is  determined  by  the  extent  of  the  class,  or  classes,  of  terms  which  are 
values  of  its  variables.  All  the  terms  which  can  be  substituted  for  x,  in 
<px,  and  'make  sense',  constitute  the  range  of  <px.  If  <px  be  'x  is  mortal', 
the  range  of  this  function  is  the  aggregate  of  all  the  individual  terms  for 
which  'x  is  mortal'  is  either  true  or  false.  Thus  the  "range  of  significance" 
is  to  propositional  functions  what  the  "universe  of  discourse"  is  to  class 
terms.  Two  propositional  functions,  <px  and  tyy,  may  be  such  that  the 
class  of  values  of  x  in  px,  or  the  range  of  <px,  is  identical  with  the  class  of 
values  of  y  in  ^y,  or  the  range  of  $y.  Or  the  two  functions  may  have 
different  ranges  of  significance,  'x  is  a  man'  and  'x  is  a  poet'  will  have  the 
same  range,  though  the  values  of  x  for  which  they  are  true  will  differ.  Any  x 
for  which  'x  is  a  man'  is  either  true  or  false,  is  also  such  that  'a:  is  a  poet' 
is  either  true  or  false.  But  some  x's  for  which  'a;  is  a  poet'  is  either  true 
or  false  are  such  that  'x  precedes  a*+l'  is  nonsense,  'a;  is  a  poet'  and 
'x  precedes  x+1'  have  different  ranges.4  It  is  important  to  note  that  the 

4  According  to  Mr.  Russell's  "theory  of  types"  (see  Principia,  i,  pp.  41-42),  the  one 
fundamental  restriction  of  the  range  of  a  propositional  function  is  the  principle  that  nothing 


234  A  Survey  of  Symbolic  Logic 

range  of  <px  is  determined,  not  by  x,  but  by  <p.  <px  and  <py  are  the  same 
function. 

If  we  have  a  prepositional  function  of  two  variables,  say  ex  is  a  citizen 
of  y ',  we  must  make  two  substitutions  in  order  to  turn  it  into  a  proposition 
which  is  either  true  or  false.  And  we  conceive  of  two  aggregates  or  classes — 
the  class  of  values  of  the  first  variable,  x,  and  the  class  of  values  of  the  second 
variable,  y.  These  two  classes  may,  for  a  given  function,  be  identical,  or 
they  may  be  different.  It  depends  upon  the  function.  "John  Jones  is  a 
citizen  of  Turkey"  is  either  true  or  false;  "Turkey  is  a  citizen  of  John 
Jones"  is  nonsense.  But  "3  precedes  5"  is  either  true  or  false,  as  is  also 
"5  precedes  3".  The  range  of  x  and  of  y  in  \f/(x,  y}  depends  upon  ^,  not 
upon  x  and  y. 

A  convenient  method  of  representing  the  values  of  x  in  <f>x  is  by  x\,  x2, 
Xz,  etc.  This  is  not  to  presume  that  the  number  of  such  values  of  x  in  <px 
is  finite,  or  even  denumerable.  Any  sort  of  tag  which  would  distinguish 
these  values  as  individual  would  serve  all  the  uses  which  we  shall  make 
of  Xi,  x2,  xz,  etc.,  equally  well.  If  Xi,  x2,  x3,  etc.,  are  individuals,5  then 
<pXi,  <px2,  (f>xs,  etc.,  will  be  propositions;  and  <pxn  will  be  a  proposition. 
<fXz  is  a  proposition  about  a  specified  individual;  <pxn  is  a  proposition  about 
'a  certain  individual'  which  is  not  specified.6  Similarly,  if  the  values  of  x 
in  \l/(x,  y}  be  x\,  x*,  xs,  etc.,  and  the  values  of  y  be  y\,  y2,  yz,  etc.,  then 
tfa>  2/3),  t(xz,  yn),  t(xm,  yn),  etc.,  are  propositions. 

We  shall  now  make  a  new  use  of  the  operators  II  and  S,  giving  them 
a  meaning  similar  to,  but  not  identical  with,  the  meaning  which  they  had 
in  Chapter  II.  To  emphasize  this  difference  in  use,  the  operators  are  here 
set  in  a  different  style  of  type.  We  shall  let  2x<px  represent  <f>Xi  +  <px2  +  <px3 
+  ...  to  as  many  terms  as  there  are  distinct  values  of  x  in  <px.  And  Ux<px 
will  represent  <pXi  x  <pxz  x  <pxz  x  .  . .  to  as  many  terms  as  there  are  distinct 
values  of  x  in  <px.  (We  have  heretofore  abbreviated  a  x6  to  a  b  or  a -b. 
But  where  prepositional  functions  are  involved,  the  form  of  expressions  is 

that  presupposes  the  function,  or  a  function  of  the  same  range,  can  be  a  value  of  the  func- 
tion. It  seems  to  us  that  there  are  other  restrictions,  not  derived  from  this,  upon  the 
range  of  a  function.  But,  fortunately,  it  is  not  necessary  to  decide  this  point  here. 

6  "Individuals"  in  the  sense  of  being  distinct  values  of  x  in  <px — which  is  the  only 
conception  of  "individual"  which  we  require. 

6  It  may  be  urged  that  ipxn  is  not  a  proposition  but  a  prepositional  function.  The 
question  is  most  difficult,  and  we  cannot  enter  upon  it.  But  this  much  may  be  said: 
Whenever,  and  in  whatever  sense,  statements  about  an  unspecified  individual  can  be 
asserted,  <pxn  is  a  proposition.  If  any  object  to  this,  we  shall  reply  "A  certain  gentle- 
man is  confused".  Peirce  has  discussed  this  question  most  acutely.  (See  above,  pp. 
93-94.) 


Systems  Based  on  Material  Implication  235 

likely  to  be  complex.  Consequently  we  shall,  in  this  chapter,  always 
write  "products"  with  the  sign  x.) 

The  fact  that  there  might  be  an  infinite  set  of  values  of  x  in  <px  does 
not  affect  the  theoretical  adequacy  of  our  definitions.  For  nothing  here 
depends  upon  the  order  of  <pxm,  <pxn,  <pxp,  and  it  is  only  required  that  the 
values  of  x  which  are  distinct  should  be  identifiable  or  "tagable".  The  ob- 
jection that  the  values  of  x  might  not  be  even  denumerable  is  more  serious, 
but  the  difficulty  may  be  met  by  a  device  to  be  mentioned  shortly. 

Since  <pxi,  <pxz,  <f>xs,  etc.,  are  propositions,  <pxi  +  <pxz  +  <?x3  +  . . .  is  a 
proposition — the  proposition,  "Either  <pxi  or  <pxz  or  <px3  or  ...  etc.".  Thus 
I,x(px  represents  "For  some  value  of  x  (at  least  one),  <px  is  true".  And 
2x<px  is  a  proposition.  Similarly,  <px\  x  <px2  x  <pxs  x  ...  is  the  joint  assertion 
of  <px\  and  tpxz  and  <px3,  etc.  Thus  Hx<f>x  represents  the  proposition  "For 
all  values  of  x,  <px  is  true".  We  may  translate  2x<px  loosely  by  "  <px  is 
sometimes  true",  and  Iix<px  loosely  by  "  <px  is  always  true".  This  trans- 
lation fails  of  literal  accuracy  inasmuch  as  the  variations  of  x  in  (px  may 
not  be  confined  to  differences  of  time. 

The  conception  of  a  prepositional  function,  <px,  and  of  the  class  of  values 
of  the  variable  in  this  function,  thus  give  us  the  new  types  of  proposition, 
<pxs,  <pxn,  2x<px,  and  Ux<px.  Since  the  laws  of  the  Two-Valued  Algebra 
hold  for  propositions  generally,  all  the  theorems  of  that  system  will  be 
true  when  propositions  such  as  the  above  are  substituted  for  a,  b,  ...  p,  q, 
etc.  (We  must,  of  course,  remember  that  while  a,  b,  ...  p,  q,  etc.,  in  the 
Two-Valued  Algebra  represent  propositions,  x  in  <px,  etc.,  is  not  a  proposi- 
tion but  a  variable  whose  values  are  individual  things.  In  the  theorems 
to  follow,  we  shall  sometimes  need  a  symbol  for  propositions  in  which  no 
variables  are  specified.  To  avoid  any  possible  confusion,  we  shall  represent 
such  propositions  by  a  capital  letter,  P.)  We  may,  then,  assume  as  already 
proved  any  theorem  which  can  be  got  by  replacing  a,  b,  ...  p,  q,  etc.,  in 
any  proposition  of  the  Two-Valued  Algebra,  by  <px3,  <f>xn,  2x<px,  or  Ilx<px. 
Additional  theorems,  which  can  be  proved  for  propositions  involving  values 
of  variables,  will  be  given  below.  These  are  to  be  proved  by  reference  to 
earlier  theorems,  in  Chapter  II  and  in  Section  I  of  this  chapter.  As  before, 
the  number  of  the  theorem  by  which  any  step  in  proof  is  taken  will  be  given 
in  square  brackets.  Since  the  previous  theorems  are  numbered  up  to  9-, 
the  additional  theorems  of  this  section  will  be  numbered  beginning  with  10  • . 

One  additional  assumption,  beyond  those  of  the  Two-Valued  Algebra, 
will  be  needed.  The  propositions  which  have  been  proved  in  sufficiently 


236  A  Survey  of  Symbolic  Logic 

general  form  to  be  used  where  sums  and  products  of  more  than  three  terms 
are  in  question  all  require  for  their  demonstration  the  principle  of  mathe- 
matical induction.  If,  then,  we  wish  to  use  those  theorems  in  the  proofs 
of  this  section,  we  are  confronted  by  the  difficulty  that  the  number  of 
values  of  x  in  <px,  and  hence  the  number  of  terms  in  ~Zx<f>x  and  Tlx<f>x  may 
not  be  finite.  And  any  use  of  mathematical  induction,  or  of  theorems 
dependent  upon  that  principle  for  proof,  will  then  be  invalid  in  this  con- 
nection. Short  of  abandoning  the  proposed  procedure,  two  alternatives 
are  open  to  us:  we  can  assume  that  the  number  of  values  of  any  variable 
in  a  prepositional  function  is  always  finite;  or  we  can  assume  that  any 
law  of  the  algebra  which  holds  whatever  finite  number  of  elements  be  involved 
holds  for  any  number  of  elements  whatever.  The  first  of  these  assumptions 
would  obviously  be  false.  But  the  second  is  true,  and  we  shall  make  it. 

This  also  resolves  our  difficulty  concerning  the  possibility  that  the 
number  of  values  of  x  in  <px  might  not  be  even  denumerable,  and  hence 
that  the  notation  <pxi  +  <px»  +  <pxs  +  .  .  .  and  tpx\  x  tpxz  x  <pxs  x  .  .  .  might  be 
inadequate.  We  can  make  the  convention  that  if  the  number  of  values  of  x 
in  any  function,  <px,  be  not  finite,  <px\  +  <px2  +  <f>x3  +  .  .  .,  or  2x<px,  and 
<f>Xi  x  <pxz  x  <px3  x  .  .  .  ,  or  nx  <px,  shall  be  so  dealt  with  that  any  theorem  to 
be  proved  will  be  demonstrated  to  hold  for  any  finite  number  of  values 
of  #  in  <px;  and  this  being  proved,  our  assumption  allows  us  to  extend  the 
theorem  to  any  case  in  which  the  values  of  the  variable  in  the  function  are 
infinite  in  number.  This  principle  will  be  satisfactorily  covered  by  the 
convention  that  <pxi  +  <px2  +  <px3  +  .  .  .  and  <px\  x  <px2  x  <pxs  x  .  .  .  shall  always 
be  supposed  to  have  a  finite  but  undetermined  number  of  terms,  and  any 
theorem  thus  proved  shall  be  presumed  independent  of  the  number  of 
distinct  values  of  any  variable,  x,  which  is  involved.7 

This  postulate,  and  the  convention  which  makes  it  operative,  will  be 
supposed  to  extend  also  to  functions  of  any  number  of  variables,  and  to 
sums,  products,  and  negatives  of  functions. 

No  further  postulates  are  required,  but  the  following  definitions  are 
needed  : 


10-01  2<px  =  2x<px  =  <f>Xi+  <px<t  +  <f>xs+  ....  Def. 
10-02  H<fX  =  Ux<px  =  <pxi  x  <pxz  x  <pxz  x  ....  Def. 
10-03  -<px  =  -{<px}.  Def. 

7  This  procedure,  though  not  invalid,  is  far  from  ideal,  as  are  many  other  details  of 
this  general  method!  We  shall  gather  the  main  criticisms  together  in  the  last  section  of 
this  chapter.  But  it  is  a  fact  that  in  spite  of  the  many  defects  of  the  method,  the  results 
which  it  gives  are  without  exception  valid. 


Systems  Based  on  Material  Implication  237 

10-031     -<pxn  =  ~{<pxn}.         Def. 
10-04     -ttx<px  =  -{Ux<f>x}.         Def. 
10-05     -2x<px  =  -{2x<px}         Def. 

The  last  four  merely  serve  to  abbreviate  the  notation. 

Elementary  theorems  concerning  propositions  which  involve  values  of 
one  variable  are  as  follows: 

10-1     ^<px  =  -U-ipx. 

[5-951]   <f>Xi+ <px2+ < 
10-12     U<px  =  -2  -<px. 

[5-95]   <pxi  x  <px2  x  (f> 

10-1  states  that  "For  some  values  of  x,  <px  is  true"  is  equivalent  to  the 
denial  of  "For  all  values  of  x,  <px  is  false".  10-12  states  that  "For  all 
values  of  x,  <px  is  true"  is  equivalent  to  the  denial  of  "For  some  values 
of  x,  <px  is  false".  These  two  represent  the  extension  of  De  Morgan's 
Theorem  to  propositions  which  involve  values  of  variables.  They  might 
be  otherwise  stated:  "It  is  true  that  all  x  is  •  •  •  "  is  equivalent  to  "It  is 
false  that  some  x  is  not  •••"',  and  "It  is  true  that  some  xis  •  •  •  "  is  equiva- 
lent to  "It  is  false  that  all  x  is  not  •  •  •  ". 

10-2     H<px  c  <pxn- 

[5  •  99]   <f>xi  x  tpxz  x  (pxs  x  .  .  .  c  tpx\ 

and  (f>Xi  x  <px2  x  <px3  x  .  .  .  c  <pxz 

and  <pxi  x  <px2  x  <px3  x  .  .  .  c  <pxa,  etc.,  etc. 

10-21      <pxnc~2(px. 

[5  •  991]  (f>Xi  c  <pxi  +  tpxz  +  <px3  +  . . . 
and  <pxz  c  <px\  +  <px2  +  <pxs  +  .  . . 
and  <pXz  c  <pxi  +  <px2  +  <px3  +  .  .  . ,  etc.,  etc. 

By  10-2,  if  <px  is  true  for  all  values  of  x,  then  it  is  true  for  any  given  value 
of  x,  or  "What  is  true  of  all  is  true  of  any  given  one".  By  10-21,  If  <?x  is 
true  for  one  given  value  of  x,  then  it  is  true  for  some  value  of  x,  or  "  What 
is  true  of  a  certain  one  is  true  of  some".  It  might  be  thought  that  the 
implication  stated  by  10-21  is  reversible.  But  we  do  not  have  2<px  c  <pxn, 
because  <pxn  may  be  <px2,  and  2  <px  c  <f>x2  would  not  hold  generally.  For 
example,  let  <px  =  "Today  (x)  is  Monday".  Then  S<px  will  mean  "Some 
day  is  Monday",  but  <pxn  will  mean  "Today  (Jan.  1)  is  Monday",  or  will 
mean  "Today  (Feb.  23)  is  Monday",  etc.  "Some  day  is  Monday"  does 


238  A  Survey  of  Symbolic  Logic 

not  imply  "Jan.  1  is  Monday",  and  does  not  imply  "Feb.  23  is  Monday  "- 
does  not  imply  that  any  one  given  day  is  Monday.     xn  in  <pxn  means  "  a 
certain  value  of  x"  in  a  sense  which  is  not  simply  equivalent  to  "some 
value  of  x".     No  translation  of  <pxn  will  give  its  exact  significance  in  this 
respect. 

10-22     H<f>xc2<px. 

[5-1,  10-2-21] 
Whatever  is  true  of  all  is  true  of  some. 

10-23     H<px  is  equivalent  to  "Whatever  value  of  x,  in  <px,  xn  may  be,  <f>xn". 

H<px  =  <pxi  x  (px2  x  <pxs  x  .  .  .  =  (<f>xi  x  <px%  x  (f>xs  x  ...  =1)  [9-01] 

And    [5-971]    <px\  x  tpx2  x  <pxs  x  .  .  .  =1    is   equivalent    to    the    set 

<f>Xi   =   1,    <pXs   =   1,    <pX3  =   1,     

And  [9-01]  <pxn  =  1  is  equivalent  to  <pxn. 

Hence  II <px  is  equivalent  to  the  set  <px\,  <px2,  <?xs,  .... 

This  proposition  is  not  tautological.  It  states  the  equivalence  of  the 
product  ipxi  x  <px2  x  <px3  x  . .  .  with  the  system  of  separate  propositions 
<f>Xi,  <px2,  <pxs,  etc.  It  is  by  virtue  of  the  possibility  of  this  proposition 
that  the  translation  of  U<px  as  "For  all  values  of  x,  <px  is  true"  is  legitimate. 
In  this  proof  we  make  use  of  the  principle,  p  =  (p  =  1) — the  only  case  in 
which  it  is  directly  required  in  the  calculus  of  propositional  functions. 

By  virtue  of  10  •  23  we  can  pass  directly  from  any  theorem  of  the  Two- 
Valued  Algebra  to  a  corresponding  theorem  of  the  calculus  of  propositional 
functions.  If  we  have,  for  example,  pc.p+q,  we  have  also  "Whatever 
value  of  x,  in  <px,  xn  may  be,  <pxnc  <f>xn  +  P".  And  hence  we  have,  by 
10-23,  TLx[<f>x  c  <px  +  P].  We  shall  later  see  the  importance  of  this:  it 
gives  us,  for  every  theorem  concerning  "material  implication",  a  cor- 
responding theorem  concerning  "formal  implication". 

Next,  we  give  various  forms  of  the  principle  by  which  any  proposition 
may  be  imported  into,  or  exported  out  of,  the  scope  of  a  II  or  2  operator. 

10-3     2<px  +  P  =  2x(<px  +  P). 

S  <pX  •*•  P  =   ( tf>Xi  +  tpX2  +  <pX3  +...)  +  P 

=  (^.r1  +  P)  +  (^2  +  P)  +  (^3  +  P)+ ...    [5-981] 


10-31     P  + 

Similar  proof. 
10  •  3  may  be  read :  " '  Either  for  some  x,  <px  is  true,  or  P  is  true '  is  equiva- 


Systems  Based  on  Material  Implication  239 

lent  to  'For  some  x,  either  <px  is  true  or  P  is  true'".  And  10-31  may  be 
read:  "'Either  P  is  true  or,  for  some  x,  <px  is  true'  is  equivalent  to  'For 
some  x,  either  P  is  true  or  <px  is  true'". 

10-32     IL<px  +  P  =  Ux(<px  +  P). 

II  <px  +  P  =  (  <pXi  x  <px2  x  <px3  x  .  .  .  )  +  P 

=  (<pxi  +  P)  x  ((facz  +  P)  x  (<pxs  +  P)  x  .  .  .   [5-941] 


10-33    P 

Similar  proof. 

''Either  P  is  true  or,  for  every  x,  <px  is  true"  is  equivalent  to  "For  every  x, 
either  P  is  true  or  <$x  is  true." 

10-34     SsOz  +  P)  =  Ss(P+  <px). 

[4-3]  Z^r  +  P  =  P+S^.T.     Hence  [10-3-31]  Q.E.D. 

10-35     nx(^r  +  P)  =  H2(P+  «*r). 
[10-32-33] 

Exactly  similar  theorems  hold  where  the  relation  of  the  two  propositions 
is  x  instead  of  +  .  The  proofs  are  so  simple  that  only  the  first  need  be 
given. 

10-36     2<px  xP  =  2x(<px  xP). 

2  <px  x  P  =  (  (pxi  +  <pxz  +  <f>xs  +  .  .  .  )  x  P 

=  (  <pxj.  x  P)  +  (  <pz2  x  P)  +  (  (pxz  x  P)  +  .  .  .   [5-94] 


"  ^  is  true  for  some  ic,  and  P  is  true",  is  equivalent  to  "For  some  x,  <px 
and  P  are  both  true". 

10-361     PxS^sr  =  Sx(Px^c). 
10-37     ft<px  xP  =  ttx(<px  xP). 
10-371 
10-38 

io-38i    nx(^.T  xP)  =  nx(P  x  <px). 

We  should  perhaps  expect  that  a  proposition,  P,  might  be  imported 
into  and  exported  out  of  the  scope  of  an  operator  when  the  relation  of  P 
to  the  other  member  of  the  expression  is  c  .  But  here  the  matter  is  not 
quite  so  simple. 


240  A  Survey  of  Symbolic  Logic 

10-4     P 


[9-3]  P  c  2v*r  =  -P  +  S<pz  =  -P 


[5-981] 
=  (P  c  tpxj  +  (P  c  <pxz)  +  (P  c  ^3)  +  •  •  •    [9-3] 


The  relation  c  ,  in  the  above,  is,  of  course,  a  material  implication. 
But  it  is  tedious  to  read  continually  "p  materially  implies  </".  We  shall, 
then,  translate  p  cq  simply  by  "p  implies  q",  or  by  "If  p,  then  q". 

10-4  reads:  "P  implies  that  for  some  .r,  <px  is  true"  is  equivalent  to 
"For  some  x,  P  implies  that  <p.r  is  true".  This  seems  clear  and  obvious, 
but  consider  the  next  : 


10-41 

[9-3]  2<pxcP  =  -S^.r  +  P  =  Il-^r  +  P  [10-12] 

=    (-tpXi  *-<pX2  X-^s  X  .  .  .)  +  P 

=  (-^d  +  P)  x  (-<p.r2  +  P)  x  (-^rs  +  P)  .  .  .  [5-941] 

P)...   [9-3] 


"'^>.T  is  true  for  some  x'  implies  P"  is  equivalent  to  "For  every  x,  <f>x 
implies  P".  It  is  easy  to  see  that  the  second  of  these  two  expressions  gives 
the  first  also  :  If  <px  always  implies  P,  then  if  <px  is  sometimes  true,  P  must 
be  true.  It  is  not  so  easy  to  see  that  2<p.r  cP  gives  Hx(<px  cP).  But  we 
can  put  it  thus:  "If  <px  is  ever  true,  then  P  is  true"  must  mean  "  <px 
always  implies  P". 


10-42     PcH^a:  =  UX(P  c  ^.r). 

[9-3]  P  c  II  <px  =  -P  +  n  <px  =  -P  +  (  <f>Xi  x  <px«  x  <pxz  x  .  .  .  ) 
=  (-P  +  <?.TI)  x  (-P  +  <pxz)  x  (-P  +  <pxs)  x  ... 

[5-941] 
=  (P  c  ipxj  x  (P  c  ^2)  x  (P  c  «p.r3)  .  .  .   [9-3] 


"P  implies  that  px  is  true  for  every  a:"  is  equivalent  to  "For  every  x,  P 
implies  <px". 


10-43     H^cP  =  2s(<pxcP). 

[9-3]  n^rcp  =  -n^  +  P  =  s-^c  +  p  [io-i] 

=    -  <.r  i  +  -  <.T2  +  -  <x3  +...+  P 


Systems  Based  on  Material  Implication  241 

=  (-  (f>xi  +  P)  +  (-  <pxz  +  P)  +  (-  <pxa  +  P)  +  ... 
[5-981] 

P)  +  ...   [9-3] 


"'<£#  is  true  for  every  a:  '.implies  that  P  is  true"  is  equivalent  to  "For 
some  x,  <px  implies  P".  At  first  sight  this  theorem  seems  to  commit  the 
"fallacy  of  division"  going  one  way,  and  the  "fallacy  of  composition" 
going  the  other.  It  suggests  the  ancient  example  about  the  separate  hairs 
and  baldness.  Suppose  <f>x  be  "  If  a:  is  a  hair  of  Mr.  Blank's,  x  has  fallen 
out".  And  let  P  be  "Mr.  Blank  is  bald".  Then  n^x  cP  will  represent 
"If  all  of  Mr.  Blank's  hairs  have  fallen  out,  then  Mr.  Blank  is  bald". 
And  Sz(  <px  c  P)  will  represent  "  There  is  some  hair  of  Mr.  Blank's  such 
that  if  this  hair  has  fallen  out,  Mr.  Blank  is  bald".  In  this  example, 
U(px  c  P  is  obviously  true,  but  2x(<px  c  P)  is  dubious,  and  their  equivalence 
seems  likewise  doubtful.  The  explanation  of  the  equivalence  is  this:  we 
here  deal  with  material  implication,  and  <pxn  c  P  means  simply  "  It  is 
false  that  (<pxn  is  true  but  P  is  false)".  U<px  cP  means,  in  this  example, 
"  It  is  false  that  all  Mr.  Blank's  hairs  have  fallen  out  but  Mr.  Blank  is  not 
bald";  and  Sx(^xcP)  means  "There  is  some  one  of.  Mr.  Blank's  hairs 
such  that  'This  hair  has  fallen  out  but  Mr.  Blank  is  not  bald'  is  false". 
No  necessary  connection  is  predicated  between  the  falling  out  of  any  single 
hair  and  baldness  —  material  implication  is  not  that  type  of  relation. 

If  we  compare  the  last  four  theorems,  we  observe  that  an  operator  in 
the  consequent  of  an  implication  is  not  changed  by  being  extended  in  scope 
to  include  the  whole  relation,  but  an  operator  in  the  antecedent  is  changed 
from  II  to  2,  from  S  to  II  .  This  is  due  to  the  fact  that  p  c  q  is  equivalent 
to  -p  +  q,  where  the  sign  of  the  antecedent  changes  but  the  consequent 
remains  the  same;  and  to  the  law  -II  Q  =  S-Q,  -S()  =  II-  Q. 

The  above  principles,  connecting  any  proposition,  P,  with  a  preposi- 
tional function  and  its  operator,  are  much  used  in  later  proofs.  In  fact, 
all  the  proofs  can  be  carried  out  simply  by  the  various  forms  of  this  principle 
and  theorems  10-1-10-23.  Since  P,  in  the  above,  may  be  any  propo- 
sition, fan,  "2  fa,  Ufa,  etc.,  can  be  substituted  for  P  in  these  theorems. 

(<px  +  fa)  and  (<px  xfa)  are,  of  course,  functions  of  x.  In  order  that 
(  (f>x  +  fa)  be  significant,  <px  must  be  significant  and  fa  must  be  significant, 
and  it  is  further  requisite  that  "Either  <px  or  fa"  have  meaning.  Such 
considerations  determine  the  range  of  significance  of  complex  functions 

like  (  (px  +  fa)  and  (  «p.r  x  fa)  .     A  value  of  x  in  such  a  function  must  be  at 

17 


242  A  Survey  of  Symbolic  Logic 

once  a  value  of  x  in  <px  and  a  value  of  x  in  fa:  xn  in  <pxn  and  in  fan,  in 
(<pxn  +  fan),  denotes  identically  the  same  individual. 

10-5     2  <px  +  2  fa  =  2Z(  <pz  +  #r). 

Since  addition  is  associative  and  commutative, 

S  <p£ 


"Either  for  some  x,  <px,  or  for  some  x,  fa"  is  equivalent  to  "For  some  x, 
either  <px  or  fa". 

If  it  be  supposed  that  the  functions,  <px  and  fa,  may  have  different 
ranges  —  i.  e.,  that  the  use  of  the  same  letter  for  the  variable  is  not  indicative 
of  the  range  —  then  S<pz  +  I,  fa  might  have  meaning  when  ^,x(tpx  +  fa)  did 
not.  But  in  such  a  case  the  proposition  which  states  their  equivalence 
will  not  have  meaning.  We  shall  make  the  convention  that  xn  in  <pxn 
and  xn  in  fan  are  identical,  not  only  in  ((pxn+fan)  and  (<pxn*fan),  but 
wherever  tpx  and  fa  are  connected,  as  in  ^<px  +  2  fa.  Where  there  is  no 
such  presumption,  it  is  always  possible  to  use  different  letters  for  the 
variable,  as  S  <px  +  ^y.  But  even  without  this  convention,  the  above 
theorem  wrill  always  be  true  when  it  is  significant  —  i.  e.,  it  is  never  false  — 
and  a  similar  remark  applies  to  the  other  theorems  of  this  section. 

10-51     U<px  xUfa  =  TLx(tpx  xfa). 

Since   x  is  associative  and  commutative,  similar  proof. 
We  might  expect  2<f>xx2fa  =  2x(<pxxfa)  to  hold,  but  it  does  not. 
"For  some  x,  x  is  ugly,  and  for  some  x,  x  is  beautiful",  is  not  equivalent  to, 
"For  some  x,  x  is  ugly  and  x  is  beautiful".     Instead  of  an  equivalence,  we 
have  an  implication: 

10  •  52     2X(  <px  x  fa)  c  2  <px  x  I,  fa. 


[5-21  (  <pxn  x  fan)  c  <pxn,  and  (  <pxn  x  fan)  c  fan 

Hence  [5-31]  2x(</>x  xfa)  cS^.r,  and  ^f((px  xfa)  cZfa 

Hence  [5  •  34]  2Z(  <px  x  fa)  cZpx  xZfa 

Similarly,  Hpx  +  Ufa  =  Ux((f>x  +  fa)  fails  to  hold.  "Either  for  every 
x,  x  is  ugly,  or  for  every  x,  x  is  beautiful  ",  is  not  equivalent  to,  "  For  every  x, 
either  x  is  ugly  or  x  is  beautiful  ".  Some  x's  may  be  ugly  and  others  beauti- 
ful. But  we  have: 


Systems  Based  on  Material  Implication  243 

10-53 


[5-21]   (pxn  c  (  <f>xn  +  fan),  and  ^.rn  c  (  <pxn  +  fe) 
Hence  [5-3]  U<f>x  c  nx(<^x  +  #r),  and  II  i/^  c  III(^a;  +  fa) 
Hence  [5  •  33]  II  <px  +  n^a:  c  nx(  <px  +  #r) 


In  the  proof  of  the  last  two  theorems,  we  write  a  lemma  for  <pxn  instead 
of  writing  it  for  <pxi,  for  <px2,  for  <px3,  etc.  For  example,  in  10-52  we  write 
(<pxn  xfan)  c  (f>xn,  instead  of  writing 


fai  c  <pxi 
(  <px2  x  fa  2)  c  <px2 
(<px3  xfa3)  c  (pxz,  etc.,  etc. 

The  proofs  are  somewhat  more  obvious  with  this  explanation.  This  method 
of  writing  such  lemmas  will  be  continued. 

With  two  prepositional  functions,  <px  and  fa,  we  can  form  two  impli- 
cation relations,  2x(<px  cfa)  and  ILx(<px  cfa).  But  2x(<px  c  fa)  states 
only  that  there  is  a  value  of  x  for  which  either  px  is  false  or  fa  is  true: 
and  this  relation  conveys  so  little  information  that  it  is  hardly  worth  while 
to  study  its  properties. 

nx(<pxcfa)  is  the  relation  of  "formal  implication"  —  "For  every  xr 
at  least  one  of  the  two,  '  <px  is  false'  and  'fa  is  true',  is  a  true  statement". 
The  negative  of  Rx(<px  cfa)  is  2x((f>x  x-fa),  so  that  H.x(<px  cfa)  may  also- 
be  read  "It  is  false  that  there  is  any  x  such  that  px  is  true  and  fa  false". 
The  material  implication,  p  c  q,  states  only  "At  least  one  of  the  two,  '  p  is 
false'  and  'q  is  true',  is  a  true  statement";  or,  "It  is  false  that  p  is  true 
and  q  false".  The  material  implication,  <pxncfan,  states  only  "At  least 
one  of  the  two,  '  <p  is  false  of  xn'  and  '  \f/  is  true  of  xn',  is  a  true  statement"; 
or  "It  is  false  that  <pxn  is  true  and  fan  is  false".  But  the  formal  impli- 
cation, Hx(<px  cfa),  states  that  however  xn  be  chosen,  it  is  false  that  <pxn 
is  true  and  fan  is  false  —  in  the  whole  range  of  <px  and  fa,  there  is  not  a 
case  in  which  <px  is  true  and  fa  false.  To  put  it  another  way,  Ux(<px  c  fa) 
means  "Whatever  has  the  predicate  <p  has  also  the  predicate  \[/". 

This  relation  has  more  resemblance  to  the  ordinary  meaning  of  "im- 
plies" than  material  implication  has.  But  formal  implication,  it  should 
be  remembered,  is  simply  a  class  or  aggregate  of  material  implications; 
Hx(<px  c  fa)  is  simply  the  joint  assertion  of  <pxt  c  fai,  <pxz  c  fa«,  <px*  c  fa^, 
etc.,  where  each  separate  assertion  is  a  material  implication.8 

8  The  whole  question  of  material  implication,  formal  implication,  and  the  usual  mean- 
ing of  "implies",  is  discussed  in  Section  v  of  Chap.  v. 


244  A  Survey  of  Symbolic  Logic 

The  properties  of  formal  implication  are  especially  important,  because 
upon  this  relation  are  based  certain  derivatives  in  the  calculus  of  classes 
and  in  the  calculus  of  relations. 


10-6     Hx(<px  cfa)  =  Hx(-<f>x  +  fa)  =  Hx 

[9  •  3]     <f>Xn  C  fan   =   -  <pXn  +  fan   =  ~(<pXn  X  -fan) 

Hence  [10-23]  Q  E.D. 

10-61       Ux((f>X  Cfa)  C  (<pXnCfan). 

[10-2] 
If  (?x  formally  implies  fa,  then  <pxn  materially  implies  fan. 

10-611     [Ux((f>x  cfa)  x  (pxn]  cfan. 
[9-4,  10-61] 

If  <px  formally  implies  fa  and  <p  is  true  of  xn,  then  ^  is  true  of  xn.  This  is 
one  form  of  the  syllogism  in  Barbara:  for  example,  "If  for  every  x,  'x  is  a 
man'  implies  'a;  is  a  mortal',  and  Socrates  is  a  man,  then  Socrates  is  a 
mortal". 

10-62     ILx(<f>x  cfa)  c.*2x(vx  cfa). 
[10-22] 


10-63     U 

[10-61]  If  Hx(<px  cfa),  then  <pxncfan 
Hence  [5-3]  Q.E.D. 

10-631     [Ux(  <px  c  fa)  x  n  tpx]  c  Ufa. 

[9-4,  10-62] 
If  <px  always  implies  fa  and  <px  is  always  true,  then  fa  is  always  true. 

10  •  64     Ux(  <px  c  fa)  c(2<pxc  S  fa)  . 

[10-61,  5-31] 
10  •  641     [nx(  <f>x  cfa)*?  tpx]  c  2  fa. 

[9-4,  10-64] 

If  <px  always  implies  fa  and  <px  is  sometimes  true,  then  fa  is  sometimes  true. 
10-65     [Ux((f>xcfa)  *TLx(fa  cfcr)]  cUx(ifXE  c£x). 

[10-61]    If   Ux(<pxcfa)    and   H.x(fa  c  £x),   then    <pxrcfan  and 

fan  c  £xn. 

Hence  [5-1]  whatever  value  of  x,  xn  may  be,  </>xnc£xn. 
Hence  [10-23]  nx(^arcfa;) 


Systems  Based  on  Material  Implication  245 

This  theorem  states  that  formal  implication  is  a  transitive  relation.  It  is 
another  form  of  the  syllogism  in  Barbara.  For  example  let  <px  =  'x  is  a 
Greek',  fa  =  'x  is  a  man',  and  far  =  'a*  is  a  mortal';  10-65  will  then  read: 
"If  for  every  x,  'a;  is  a  Greek'  implies  'x  is  a  man',  and  for  every  x,  'x  is  a 
man'  implies  'a:  is  a  mortal',  then  for  every  x,  'a;  is  a  Greek'  implies  'a:  is  a 
mortal'". 

10-65  may  also  be  given  the  form: 

10-651     Ux(<pxcfa)  c[Ilx(fac£x)  c  Ux(<f>x  cfar)]. 

[9-4,  10-65] 
10-652     Ur(\f/x  cfar)  c[Ux(<px  cfa)  cllx(<px  cfa;)]. 

[9-4,  10-65] 
10  -66     Ux(  <px  c  fa)  =  Ilx(-fa  c-<px). 

[3-1]  ( <pxn  c  fan)  =  (-fan  c  -  <pxn) 
Hence  [2-2,  5-3]  Q.E.D. 

Any  further  theorems  concerning  formal  implication  can  be  derived 
from  the  foregoing. 

"Formal  equivalence"  is  reciprocal  formal  implication,  just  as  material 
equivalence  is  reciprocal  material  implication.  The  properties  of  formal 
equivalence  follow  immediately  from  those  of  formal  implication. 

10-67     Hx(<f>x  =  fa)  =  [Hx(<px  c  fa)  *Hx(fa  c  «pa:)]. 

Whatever  value  of  x,  xn  may  be,  [2-2]  <pxn  =  \f/xn  is  equivalent 
to  the  pair,  <pxn  c  \{/xn  and  4>xn  c  <pxn. 
Hence  [10-23]  Q.E.D. 

10-68     [TLx(tpx  =  $x}  xnx(^  =  fa;)]  cllx(<f>x  =  fa;). 

Whatever  value  of  x,  xn  may  be,  if  <px  =  ^x  and  \px  =  fa;,  then 
<px  =  far.     Hence  [10-23]  Q.E.D. 

10-681     Ux(<f>x  =  \l/x)  c[Ux(\lfX  =  fa;)  cTLx(<px  =  fa;)]. 

[10-68,  9-4] 
10-682     Ilx(tx  =  fa;)  c[Ux(<px  =  $x)  cTLx(<px  =  fa;)]. 

[10-68,  9-4] 

Formal  equivalence,  as  indicated  by  the  last  three  theorems,  is  a  transitive 
relation. 

10-69     Ux(<px  =  \l/x)  c(<f>xn  =  fan);     TLx(<px  =  \f/x)  c(JI<f>x  =  H^a:);     and 
Hx(<px  =  fa)  c(2<px  =  2 fa). 
[2-2,  10-61-62- 63] 


246  A  Survey  of  Symbolic  Logic 

10-691 


[3-2,  10-23] 

If  we  wish  to  investigate  the  propositions  which  can  be  formed  from 
functions  of  the  type  of  (<px  x^y)  and  (<px  +  \f/y),  where  the  range  of  sig- 
nificance of  <px  may  differ  from  that  of  ^y,  we  find  that  these  will  involve 
two  operators  —  SJIj/^r;  \f/y),  Ily2x(<px;  \l/y],  etc.  And  these  are  special 
cases  of  a  function  of  two  variables,  (tpxx^y)  and  (ipx  +  \f/y)  are  special 
cases  of  £(x,  y).  Hence  we  must  first  investigate  functions  of  two  variables 
in  general. 

III.     PROPOSITIONAL  FUNCTIONS  OF  Two  OR  MORE  VARIABLES 

A  prepositional  function  of  two  variables,  <p(x,  y),  gives  the  derivative 
propositions  <p(xm,  yn},  ILx<p(x,  yn),  2x2y<f>(x,  y),  2vnx<p(x,  y),  etc.  The 
range  of  significance  of  <p(x,  y)  will  comprise  all  the  pairs  (x,  y)  such  that 
<p(x,  y}  is  either  true  or  false.  We  here  conceive  of  a  class  of  individuals, 
#1,  #2,  xs,  etc.,  and  a  class  of  individuals,  yi,  y^,  ys,  etc.,  such  that  for  any 
one  of  the  x's  and  any  one  of  the  y's,  <p(x,  y)  is  either  true  or  false. 

As  has  already  been  pointed  out,  the  function  may  be  such  that  the 
class  of  values  of  x  is  the  same  as  the  class  of  values  of  y,  or  the  values  of  x 
may  be  distinct  from  the  values  of  y.  If,  for  example,  <p(x,  y)  be  "x  is 
brother  of  y",  the  class  of  x's  for  which  <p(x,  y)  is  significant  consists  of 
identically  the  same  members  as  the  class  of  y's  for  which  <p(x,  y)  is  sig- 
nificant.9 In  such  a  case,  the  range  of  significance  of  <p(x,  y)  is  the  class  of 
all  the  ordered  couples  which  can  be  formed  by  combining  any  member  of 
the  class  with  itself  or  with  any  other.  Thus  if  the  members  of  such  a 
class  be  a\,  a^,  a3,  etc.,  the  class  of  couples  in  question  will  be10 

(ai,  o.i),  (ai,  rz2),  («i,  a3),  •  •  • 

(a2,  ai),  (a2,  a,),  (02,  a8),  ... 

(o3,  ai),   (a3,  a2),  (a3,  a3),  ... 
.  .  .    Etc.,  etc. 

But  if  <p(x,  y)  represent  "a:  is  a  citizen  of  y",  or  "a:  is  a  proposition  about 
y",  or  "x  is  a  member  of  the  class  y",  the  class  of  x's  and  the  class  of  y's 
for  which  <p(x,  y)  is  significant  will  be  mutually  exclusive. 

9  Presuming  that  "A  is  brother  of  A  "  is  significant  —  i.  e.,  false. 

10  Schroder  treats  all  relatives  as  derived  from  such  a  class  of  ordered  couples.     (See 
Alg.  Log.,  in,  first  three  chapters.)     But  this  is  an  unnecessary  restriction  of  the  logic  of 
relatives. 


Systems  Based  on  Material  Implication  247 

Although  <f>(x,  y)  represents  some  relation  of  x  and  y,  it  does  not  neces- 
sarily represent  any  relation  of  the  algebra,  such  as  x  cy  or  x  =  y;  and  it 
cannot  represent  relations  which  are  not  assertable. 

<P(XI,  y),  <p(x*,  y),  etc.,  are  prepositional  functions  of  one  variable,  y. 
Hence  Uy<f>(xi,  y),.Uy<p(x^,  y),  2y<p(xi,  y),  etc.,  are  propositions,  the  meaning 
and  properties  of  which  follow  from  preceding  definitions  and  theorems. 
And  H.y<f>(x,  y),  2y<p(x,  y},  IIx<p(x,  y},  and  2x<p(x,  y)  are  prepositional 
functions  of  one  variable.  We  can,  then,  define  propositions  involving 
two  variables  and  two  operators,  as  follows: 

11-01  njI^Gr,  y)  =  Ux{Us<p(x,  y)}.  Def. 

11-02  ZJI^Gc,  T/)  =  2x{nv<p(x,  y}}.  Def. 

1  1  •  03  nzSy  <p(x,  y)  =  IT*  {  2,  <p(x,  y)  }  .  Def. 

1  1  •  04  SxStf  <p(x,  y}  =  Zx  {  S,  v(x,  y}  }  .  Def. 

It  will  be  seen  from  these  definitions  that  our  explanation  of  the  range 
of  significance  of  functions  of  twro  variables  was  not  strictly  required;  it 
follows  from  the  explanation  for  functions  of  one  variable.  The  same  con- 
vention regarding  the  number  of  values  of  variables  and  interpretation  of 
the  propositions  is  also  extended  from  the  theory  of  functions  of  one 
variable  to  the  theory  of  functions  of  two. 

(Where  the  first  variable  has  a  subscript,  the  comma  between  the  two 
will  be  omitted:  <p(xjy)  is  <p(xz,  y),  etc.) 

Since  Hy<p(x,  y)  is  a  prepositional  function  of  one  variable,  x,  the  defini- 
tion, 10-02,  gives  us 

UzUy<p(x,  y)  =  Ux{Uy<p(x,  y}}  =  U.y<p(x<y)  xUy<p(x*y)  xUy<p(xsy)  x  .  .  . 
And  the  expansion  of  this  last  expression,  again  by  10-02,  is 


x  {  <p(x2yi)  x  <p(xtyd  x  ^(x^ys)  x  .  .  . 
x  {  <p(x$yi)  x  <p(x3y»)  x  <p(xsys)  x  .  .  . 
x  ...  Etc.,  etc. 

And  similarly,  by  10-01, 

2xUy<p(x,  y)  =  2t{ny<p(x,  y)}  =  Uy<f>(xiy)  +  tty<p(xzy 
And  the  expansion  of  the  last  expression,  by  10  •  02,  is 


x  <pxiy  x  <pxiy  x 
+  {  <p(x2yi)  x  <p(xzyi}  x  ^(^2^/3)  x 
+  {  <p(x3yi)  x  v(x3y2)  x  <p(xsyz)  x 
+  .  .  .  Etc.,  etc. 


248  A  Survey  of  Symbolic  Logic 

Or,  in  general,  any  prepositional  function  with  two  operators  is  expanded 
into  a  two-dimensional  array  of  propositions  as  follows: 

(1)  The  operator  nearest  the  function  indicates  the  relation  (+   or   x) 
between  the  constituents  in  each  line. 

(2)  The  subscript  of   the  operator  nearest  the  function  indicates  the 
letter  which  varies  within  the  lines. 

(3)  The  operator  to  the  left  indicates  the  relation  (  +   or    x  )  between 
each  two  lines. 

(4)  The  subscript  of  the  operator  to  the  left  indicates  the  letter  which 
varies  from  line  to  line. 

Some  caution  must  be  exercised  in  interpreting  such  propositions  as 
2x1Iv<p(x,  y),  etc.  It  is  usually  sufficient  to  read  S^IIj,  "For  some  x  and 
every  y",  but  strictly  it  should  be  "For  some  x,  every  y  is  such  that". 
Thus  2xIIj,  <£>(#,  y}  should  be  "For  some  x,  every  y  is  such  that  <p(x,  y}  is 
true".  And  Hy2x<f>(x,  y)  should  be  "For  every  y,  some  x  is  such  that 
<p(x,y)  is  true".  The  two  here  chosen  illustrate  the  necessity  of  caution, 
which  may  be  made  clear  as  follows: 


That  is,  2xILy<f>(x,  y}  means  "Either  for  x\  and  every  y,  <p(x,  y}  is  true, 
or  for  xi  and  every  y,  <p(x,  y)  is  true,  or  for  xs  and  every  y,  <p(x,  y)  is  true, 
...  or  for  some  other  particular  x  and  every  y,  <p(x,  y)  is  true".  On  the 
other  hand, 

Uy2x<p(x,  y}  =  ?x<p(x,  y^  xZI<P(.r,  ?/2)  xSx<p(.r,  7/3)  *  .  .  . 

That  is,  Uy'Zx^x,  y}  means  "For  some  x  and  yi,  <p(x,  i/)  is  true,  and  for 
some  x  and  y?,  <p(x,  y)  is  true,  and  for  some  x  and  ys,  <p(x,  y)  is  true,  and 
.  .  .";  or  "Given  any  y,  there  is  one  x  (at  least)  such  that  <p(x,  ?/)  is  true". 
The  following  illustration  of  the  difference  of  these  twyo  is  given  in  Principia 
Mathematical  u  Let  <p(x,  y}  be  the  prepositional  function  "If  y  is  a  proper 
fraction,  then  x  is  a  proper  fraction  greater  than  y".  Then  for  all  values 
of  y,  we  have  2x<p(x,  yn),  so  that  ~n.yZ,x<p(x,  y)  is  satisfied.  In  fact,  H^x(p(x,y) 
expresses  the  proposition:  "If  y  is  a  proper  fraction,  then  there  is  always 
a  proper  fraction  greater  than  y".  But  2xIIv<f>(x,  y)  expresses  the  propo- 
sition: "There  is  a  proper  fraction  which  is  greater  than  any  proper  frac- 
tion ",  which  is  false. 

In  this  example,  if  we  should  read  SJTy  "For  some  x  and  every  y"; 

11  See  i,  p.  161. 


Systems  Based  on  Material  Implication  249 

IlySi  "For  every  y  and  some  x",  we  should  make  equivalent  these  two 
very  different  propositions.  But  cases  where  this  caution  is  required  are 
infrequent,  as  we  shall  see. 

Where  both  operators  are  II  or  both  S,  the  two-dimensional  array  of 
propositions  can  be  turned  into  a  one-dimensional  array,  since  every  rela- 
tion throughout  wrill  be  in  the  one  case  x  ,  in  the  other  +  ,  and  both  of 
these  are  associative  and  commutative.  It  follows  from  our  discussion  of 
the  range  of  significance  of  a  function  of  two  variables  that  any  such  func- 
tion, <p(x,  y),  may  be  treated  as  a  function  of  the  single  variable,  the  ordered 
couple,  (x,  y).  Hence  we  can  make  the  further  conventions: 

11-05  2x2y<p(x,  y)  =  S(x,  y}<p(x,  y)  =  Sx,  y<p(x,  y). 
11-06  Uxny<p(x,  y)  =  n(l>  V)<f>(x,  y)  =  Iix,  y<p(x,  y}. 
The  second  half  of  each  of  these  serves  merely  to  simplify  notation. 

11-07  If  xr  and  ys  be  any  values  of  x  and  y,  respectively,  in  <p(x,  y),  there 
is  a  value  of  (x,  y)  —  say,  (x,  y)n  —  such  that  v(x,  y)n  =  v(xrys}. 

11-05  and  11-06  could  be  derived  from  11-07,  but  the  process  is  tedious, 
and  since  our  interest  in  such  a  derivation  would  be  purely  incidental, 
we  prefer  to  set  down  all  three  as  assumptions. 

If  we  wish  to  identify  a  given  constituent  of  SZ)  y<p(x,  y)  with  a  con- 
stituent of  2x'2y(p(x,  y),  some  convention  of  the  order  of  terms  in  Sz>  y<p(x,  y) 
is  required,  because  if  the  order  of  constituents  in  ^x'2y(p(x,  y}  be  unaltered, 
this  identification  will  be  impossible  unless  the  number  of  values  of  y  is 
determined  —  which,  by  our  convention,  need  not  be  the  case.  Hence  we 
make,  concerning  the  order  of  terms  in  Sz,  y<p(x,  y),  the  following  conven- 
tion :  (f(xmyn)  precedes  <p(xrys)  ifm  +  ra<r+«,  and  where  m  +  n  =  r  +  s, 
if  n  <  s.  Thus  the  order  of  terms  in  2X,  y<p(x,  y}  will  be 


This  arrangement  determines  an  order  independent  of  the  number  of  values 
of  x,  or  of  y,  so  that  the  equivalent  of  <p(x,  y)n  in  terms  of  <p(xrys]  can  always 
be  specified.12  An  exactly  similar  convention  is  supposed  to  govern  the 
arrangement  of  terms  in  11*,  y<p(x,  y)  and  their  identification  with  the  terms 
of  Hxlly<p(x;  y}.  These  conventions  of  order  are  never  required  in  the 
proof  of  theorems:  we  note  them  here  only  to  obviate  any  theoretical 

12  This  arrangement  turns  the  two-dimensional  array  into  a  one-dimensional  by  the 
familiar  device  for  denumerating  the  rationals  —  i.  e.,  by  proceeding  along  successive  di- 
agonals, beginning  with  the  upper  left-hand  corner. 


250  A  Survey  of  Symbolic  Logic 

objection.  The  identification  of  2x2v<f>(x,  y)  with  2X,  v<p(x,  y),  and  of 
H.xllv(p(x,  y)  with  nx,  v<f>(x,  y),  is  of  little  consequence  for  the  theory  of 
prepositional  functions  itself,  but  it  will  be  of  some  importance  in  the  theory 
of  relations  which  is  to  be  derived  from  the  theory  of  functions  of  two  or 
more  variables. 

Having  now  somewhat  tediously  cleared  the  ground,  we  may  proceed 
to  the  proof  of  theorems.  Since  2X)  y<p(x,  y)  and  Hx,  y<p(x,  y)  may  be  re- 
garded as  involving  only  one  variable,  (x,  y),  many  theorems  here  follow 
at  once  from  those  of  the  preceding  section. 

11-1     Sx,  y<p(x,  y)  =  2t2v<p(x,  y)  =  -HZf  v-<p(x,  y)  =  -{UtUv-<p(x,  y)}. 

[11-05-06,  10-05] 
11-12     Hx,  v<p(x,  y}  =  nxUy<p(x,  y}  =  -SXf  v-<p(x,  y}  =  -{^x^y-<p(x,  y)\. 

[11-05-06,  10-04] 

11-2     nx,  y<p(x,  y)  c  <f>(x,  y)n- 
[10-2] 

11-21      v(x,y}nc*Lx,y<p(x,y}. 

[10-21] 
11-22     n*f  y<p(x,  y)  c  SXf  y  <p(x,  y) . 

[10-22] 

11-23  nx,  y<p(x,  y}  is  equivalent  to  "Whatever  value  of  (x,  y},  in  <p(x,  y), 
(x,  y)n  may  be,  <p(x,  y)n". 

[10-23] 

11-24     Uxllv<p(x,  y)  is  equivalent  to  "Whatever  values  of   x  and  y,  in 

<p(x,  y),  x,  and  y,  may  be,  <p(xryty. 

[10-23]  njlv<p(£,  y)  is  equivalent  to  "Whatever  value  of  x,  in 
Hv<f>(x,  y),  xr  may  be,  Uy<f>(xry)".  And  Hy<p(xry)  is  equivalent  to 
"Whatever  value  of  y,  in  <p(x,y),  ys  may  be,  <p(xrys}".  But  [11-01] 
the  values  of  x  in  T\y(p(xry)  are  the  values  of  x  in  <p(x,  y).  Hence 
Q.E.D. 

11-25  "Whatever  value  of  (x,y),  in  <p(x,y),  (x,y}n  may  be,  <p(x,  y)n" 
is  equivalent  to  "Whatever  values  of  x  and  y,  in  <p(x,  y),  xr  and  ys  may  be, 
v(xry.)  "• 

[11-06-23-24] 

1 1  •  26     nxlly  <p(x,  y)  c  Uy  <p(xny) . 
[11-01,  10-2] 


Systems  Based  on  Material  Implication  251 

11-27     UxUy<p(x,  y}  =  HvUx<p(x,  y). 

Since  x  is  associative  and  commutative,  Q.E.D. 
11-28  2X2^(X  y)  =  2v2x<p(x,  y}. 

Since  +  is  associative  and  commutative,  Q.E.D. 
1  1  •  29  Uxlly  <p(x,  y}  c  nx  <f>(x,  yn)  . 

[11-26-27] 

11-291     Uxny<p(x,  y}  c  <p(xryt). 
[2-2,  11-24] 

11-3     IIJIj,  <p(x,  y)  c  2xntf  <p(x,  y}  . 

[11-01,  10-21] 
11-31     2JI,  <p(x,  y}  c  n,2x  p(ar,  y)  . 

[11-03]    UyZx<p(x,  y)  = 


x 

x  ...   Etc.,  etc. 

Since  x  is  distributive  with  reference  to  +  ;  this  expression  is  equal 
to  the  sum  of  the  products  of  each  column  separately,  plus  the  sum 
of  all  the  cross-products,  that  is,  to 

A  +  {  vfayi)  x  <p(xiyz)  x  <p(xiy3)  x  .  .  .  } 
+  {  <p(x2yi~)  x  (f>(x2yz)  x 
+  {  ^(ar3yi)  x  <p(xsy2)  x 
+  .  .  .   Etc.,  etc. 

where  A  is  the  sum  of  all  cross-products. 
But  [11-02]  this  is  2xUv<p(x,  y)  +  A. 
Hence  2xlly<p(x,  y}  +  A  =  ILy2x<p(x,  y). 
Hence  [5-21]  2xny<p(x,  y}  cUy^lV(x,  y). 

We  have  already  called  attention  to  the  fact  that  the  implication  of  11-31 
is  not  reversible  —  that  2xlly<f>(x,  y}  and  Hy2x<p(x,  y)  are  not  equivalent. 

11-32     Ux2y  <p(x,  y}  c  2xS^(x,  y)  . 

[11-03]  Ux2v<f>(x,  y)  = 

[11-04]  2*2^(3-,  y)  = 

And  [5-992]  ?y<p(xiy)  x2^(^22/)  x2^(.T3?/)  x 


We  have  also  the  propositions  concerning  formal  implication  where 


252  A  Survey  of  Symbolic  Logic 

functions  of  two  variables  are  concerned.  The  formal  implication  of 
t(%,  y}  by  <p(x,  y}  may  be  written  either  Ux,  v[<f>(x,  y)  c  $(x,  y}}  or 
HxHv[<f>(x,  y)  CTJ/(X,  y}].  By  11-06,  these  two  are  equivalent.  We  shall 
give  the  theorems  only  in  the  first  of  these  forms. 

11-4     Ux,  y[<p(x,  y)  c  t(x,  y)]  =  Ilx,  y[-<p(x,  y)  +  t(x,  y)] 

=  nx,  v-[<p(x,  y)  x-^(z,  y)]. 
[10-6] 

11-41     II*,  v[<p(x,  y)cj(x,  y)]c[v(x,  y}nc^(x,  ?/)„]. 

[10-61] 
11-411     {Ux,  v[<p(x,  y)  c  $(x,  y)]  x  <p(x,  y}n]  c  if,(x,  y)«. 

[10-611] 

11-42     n2,  v[<p(x,  y}  c  f(x,  y}]  c  2X)  v[<p(x,  y)  c  f(x,  y)]. 
[10-62] 

11-43     II*.  v[<p(x,  y}  c  f(x,  y}]  c  [n«,  v<p(x,  y}  c  Ux,  rf(x,  y}]. 

[10-63] 
11-431     |nlf  y[<f>(x,  y)  c$(x,  y}]  *UX,  v<p(x,  y)}  cUz,  y$(x,  y). 

[10-631] 
11-44     nx,  v[<p(x,  y}  c  f(x,  y)]  c  [Sx,  y<f>(x,  y)  c  Sx,  vt(x,  y)}. 

[10-64] 
11-441     {II*,  v[<p(x,  y}  c$(x,  y}]  x2x,  y<p(x,  y)}  c  Sx,  ^(.T,  y). 

[10-641] 
11-45     {I!*,  v[^(o:,  y)c$(x,  y)]  xll*.  tf[^(ar,  y)  cf(a 


[10-65] 

11-451     n,,  „[*(*,  y)c*(ar,y)] 

c  {nx,  v[^(a;,  y)  c^fe  y)]  cnx,  y[<t>(x,  y}  cf(x,  y)]}. 
[10-651] 

11-452     nx,v[i(x,y)ct(x,y)] 

c  {Ux,  y[<p(x,  y)ct(x,  y)]cUx,  v[<p(x,  y)  cf(x,  y)]}. 
[10-652] 

11-46     nlf  y[<p(x,  y)  c  $(x,  y}]  =  Ux,  v[-f(x,  y)  c-<p(x,  y)]. 

[10-66] 

Similarly,  we  have  the  theorems  concerning  the  formal  equivalence  of 
functions  of  two  variables. 


Systems  Based  on  Material  Implication  253 

11-47     n*.  y[v(x,  y)  =  f(x,  y}}  =  {Hx,  y[<p(x,  y)  c  f(x,  y)] 


[10-67] 
11-48     {n*,  v[<p(x,  y}  =  t(x,  y}]  *UX,  vty(x,  y}  =  £(x,  y)]} 

cns,  y[<p(x,  y)  =  $(x,  y)]. 

[10-68] 
11-481     IIx,  y[(p(x,  y)  =  $(x,  y}]  c  {H.x,  y[^(x,  y}  =  £(x,  y)] 

[10-681] 
11-482     nx>  v[$(x,  y}  =  t(x,  y)]  c  |ns,  v[<p(x,  y)  =  $(x,  y}] 

cllx,  y[<p(x,  y)  =  {(x, 
[10-682] 
11-49     n*,  y[<p(x,  y)  =  $(x,  y}]  c  [<p(x,  y)n  =  f(x,  y)n] 

c[Ux,  v<p(x,  y}  =  Ux,  v$(x,  y)] 
C[2X,  V<P(X,  y)  =  SZ|  vt(x,  y)]. 
[10-69] 

11-491     nz,  v[<p(x,  y}  =  $(x,  y}]  =  Ux,  y[-<p(x,  y}  =  -f(x,  y}]. 
[10-691] 

Further  propositions  concerning  functions  of  two  variables  are  simple 
consequences  of  the  above. 

The  method  by  which  such  functions  are  treated  readily  extends  to 
those  of  three  or  more  variables.  <p(x,  y,  z)  may  be  treated  as  a  function 
of  three  variables,  or  as  a  function  of  one  variable,  the  ordered  triad  (x,  y,  z} ; 
just  as  \]/(x,  y}  can  be  treated  as  a  function  of  x  and  y,  or  of  the  ordered 
pair  (x,  y).  Strictly,  new  definitions  are  required  with  each  extension  of 
our  theory  to  a  larger  number  of  variables,  but  the  method  of  such  extension 
will  be  entirely  obvious.  For  three  variables,  we  should  have 

SJI^II^OE,  y,  z}  =  sx{nvii,<p(a:,  y,  z)\ 

Etc.,  etc. 

It  is  interesting  to  note  that  the  most  general  form  for  the  analogues  of 
11 -05  and  11-06  will  be 

n(z,  y,  ,)  v(x,  y,  z)  =  IIJIcj,,  ,)  <p(x,  y,  z) 
and     2(x,  „,  ,)<p(x,  y,  z)  =  SzS(tff  ,)<p(x,  y,  z) 

Since  IIJI^,  ,)<p(x,  y,  z)  =  n(v,  ,)<?(&#,  2)  xn(v,  *)<p(x2y,  z)  xn(v,  ,)<f>(xty,  z) 


254  A  Survey  of  Symbolic  Logic 

x  .  .  .,  and  !!(„,  *)<f>(xny,  2)  =  nyllz<f>(xny,  z),  etc.,  we  shall  be  able  to  deduce 

n(l,  y,  *)¥>(#,  y,  2)  =  nxn(lM  z)<p(x,  y,  z)  =  n(l,  y)nz<p(x,  y,  z) 
=  UyU(x,  ,)<p(x,  y,  z)  =  UxUuUe(f>(x,  y,  2) 

And  similarly  for  2(x,  v,  z).  This  calls  our  attention  to  the  fact  that  <p(x, 
y,  z)  can  be  treated  not  only  as  a  function  of  three  variables  or  as  a  function 
of  one,  but  also  as  a  function  of  two,  x  and  (y,  2)  or  (x,  y)  and  2  or  (x,  2) 
and  y. 

In  general,  the  conventions  of  notation  being  extended  to  functions  of 
any  number  of  variables,  in  the  obvious  way,  the  analogues  of  preceding 
theorems  for  functions  of  two  will  follow. 


We  failed  to  treat  of  such  expressions  as  U<px  xrty?/,  I,<f>x  +  Il\f/y,  etc., 
under  the  head  of  functions  of  one  variable.  The  reason  for  this  omission 
was  that  such  expressions  find  their  significant  equivalents  in  propositions 
of  the  type  nxlly(<f>x  x^y},  2xny(<f>x  +  -tyy),  etc.,  and  these  are  special  cases 
of  functions  of  two  variables.  We  may  also  remind  the  reader  of  the 
difference  between  two  such  expressions  as  II  <px  +  Ity.r  and  n  <px  +  U\f/y. 
The  ranges  of  the  two  functions,  v  and  \l/,  need  not  be  identical;  there 
may  be  values  of  x  in  <px  which  are  not  values  of  y  in  -fyy.  But  in  any 
expression  of  the  form  <pxn  x  \f/xn,  xn  as  a  value  of  x  in  <px  must  be  identical 
with  xn  as  a  value  of  x  in  \f/x.  For  this  reason,  we  have  adopted  the  con- 
vention that  where  the  same  letter  is  used  for  the  variable  in  two  related 
functions,  these  functions  have  the  same  range.  Hence  the  case  where 
we  have  <px  and  \f/y  is  the  more  general  case,  in  which  the  functions  are  not 
restricted  to  the  same  range.  Theorems  involving  functions  of  this  type 
will  not  always  be  significant  for  every  choice  of  <p  and  \l/.  There  may  even 
be  cases  in  which  an  implication  is  not  significant  though  its  hypothesis  is 
significant.  But  for  whatever  functions  such  theorems  are  significant, 
they  will  be  true;  they  will  never  be  false  for  any  functions,  however  chosen. 

The  meaning  of  an  expression  such  as  2xHy(  <px  +  \f/y}  follows  from  the 
definition  of  2xTlv<p(x,  y). 


=         {  (  <f>Xl  +  ^l)  X  (  <PX\  +  ^2)  X 

+   {  (tpXt  +  ^0   X  (<px*  +  ^2)   X  (<pX2  + 

+  {  (  <px3  +  tyi)  x  (  <px3  +  tyz)  x  (<px»  +  ty*>  x  .  .  .  } 
+  .  .  .   Etc.,  etc. 

And  for  any  such  expression  with  two  operators  we  have  the  same  type  of 


Systems  Based  on  Material  Implication  255 

two-dimensional  array  as  for  a  function  of  two  variables  in  general.  The 
only  difference  is  that  here  the  function  itself  has  a  special  form,  <px  +  \{/y 
or  ipx  x  \f/y,  etc. 

12-1     U<px  *U\l/y  =  U\f/y  xU<px  =  UxILv(<px  x^y)  =  TLJly(^y  x  <px) 

=  Uynx(<px  x^y)  =  nyttx(ty  x  <px)> 

(1)  [1-3]  n<px*nty  =  n^xn^x. 

(2)  II  <px  x  Uif/y  =  (  pxi  x  <px2  *  <f>Xs  x  .  .  .  )  x  Ti-^y 

=  (<pxj.  x  Ityy)  x  Oz2  x  Ityy)  x  (^.T3  x  ityy)  x  ... 

[5-98] 


^)  x...     [10-371] 

.    [ii-oi] 

(3)  By  (2)  and  1-3, 

II  <px  x  U\l/y  =  (Tl\f/y  x 


x  ntf(^2/  x 

x  ntf(^  x  pz3)  x  .  .  .     [10-37] 

).    [ii-oi] 

(4)  Similarly,  n^  xll^a;  =  IIj/nz(i/'?/  x  ^x)  =  UyUx(<px  x^?/). 

"  ^a;  is  true  for  every  x  and  \l/y  is  true  for  every  y"  is  equivalent  to  "For 
every  x  and  every  y,  ipx  and  ^T/  are  both  true",  etc. 

12-2     2<px  +  2$y  =  ?ty  +2<px  =  SxSy(  <px  +  ty)  =  2x2y(if<y  +  <px) 


(1)  [4-3] 

(2)  I,<px 

Si/'?/)  +  .  .  . 
[5-981] 


ty)+  ...     [10-31] 
=  S,2,,(*a  +  ^).     [11-04] 
(3)  By  (2)  and  4  -3, 


...     [10-3] 
=  2^^+^).     [11-04] 
(4)  Similarly,  S^y+S^c  =  SyS^t/  +  ^.T)  =  SJ/Sx(v?.r  +  ^?/). 


256  A  Survey  of  Symbolic  Logic 

"Either  for  some  x,  <px,  or  for  some  y,  \l/y"  is  equivalent  to  "For  some 
x  and  some  y,  either  <?x  or  \l/y",  etc. 

12-3     SV?.T  x  Zjy  =  2^y  x  S^z  =  2x?v(<px  x^)  =  Vx*Lv(ty  x  ^r) 

=  2tf2x($«r  x^y)  =  SVS*(^2/  x  ^r). 

(1)  [1-3]  2<f>xx2ty  =  S^yxS^ar. 

(2)  S  #c  x  2^y  =  (<p£i  +  #r2  +  v^a  +  •  •  •  )  x  2^y 

x  2^y)  +  (^2  x  S^y)  +  (v?a:3  x  S^y)  +  .  .  . 

[5-94] 
x  $y)  +  Sv(  ^2  x  ^y)  +  Stf  (  ^3  x  ty) 

+  ...     [10-361] 
^).     [11-04] 

(3)  By  (2)  and  1-3, 

x  2^7/  =  (Z^?/  x  ^i)  +  (S^y  x  <px2)  +  (S^i/  x 


.  .  .     [10-36] 
.     [11-04] 
(4)  Similarly,  2^1/xS^a:  =  ^y^x(^y  x  ^)  =  SyS^C^x  x^?/). 

"For  some  x,  <pz,  and  for  some  i/,  ^?/"  is  equivalent  to  "For  some  z  and 
some  y,  px  and  ty",  etc. 

12-4 


(1)  [4-3] 

(2)  U<f>x  +  U\j/y  =  (<pxi  x  <pxz  x  (f>x3  x  .  .  .  )  + 

x  (<px3  +  n^y)  x  ... 
[5-941] 


x...     [10-371] 

[11-01] 
(3)  By  (2)  and  4  -3, 

Il<px  +  n^y  =  (Uty  +  <pxd  x  (Uty  +  <pxd 

+       3   x  ... 


...     [10-37] 
«r).     [11-01] 
(4)  Similarly,  n^y  +  n^a:  =  n1/III(i/'?/  +  <^.r)  =  nyllx(^a:  +  ^). 

"Either  for  every  x,  <px,  or  for  every  y,  \f?y"  is  equivalent  to  "For  every  x 


Systems  Based  on  Material  Implication  257 

and  every  y,  either  px  or  \fsy",  etc.  At  first  glance  this  theorem  may  seem 
invalid.  One  may  say:  "Suppose  <px  be  'If  a-  is  a  number,  it  is  odd', 
and  $y  be  'If  y  is  a  number,  it  is  even'.  Then  U<f>x  +  H\l/y  will  be  'Either- 
every  number  is  odd  or  every  number  is  even',  but  Hxlly((px  +  \l/y}  will  be 
'Every  number  is  either  odd  or  even'".  The  mistake  of  this  supposed 
illustration  lies  in  misreading  Ilxlly(<f>x  +  \l/y~).  It  is  legitimate  to  choose, 
as  in  this  case,  tpx  and  \l/y  such  that  their  range  is  identical:  but  it  is  not 
legitimate  to  read  Uxlly(<px  +  \l/y]  as  if  each  given  value  of  x  were  connected 
with  a  corresponding  value  of  y.  To  put  it  another  way:  HxTLx(<px  +  \l/x), 
as  a  special  case  of  Uxny(<px  +  -^y),  would  not  be  "For  every  value  of  x, 
either  <px  or  \fsx",  but  would  be  "For  any  two  values  of  x,  or  for  any  value 
of  x  and  itself,  either  <px  or  \f/x".  Thus  TLxTI.y(<px  +  \f/y)  in  the  supposed 
illustration  would  not  be  as  above,  but  is  in  fact  "For  any  pair  of  numbers, 
or  for  any  number  and  itself,  either  one  is  odd  or  the  other  is  even"  —  so 
that  Hpx  +  TL\f/y  and  ILJI^tpx  +  \f/y)  would  here  both  be  false,  and  are  equiva- 
lent. 

A  somewhat  similar  caution  applies  to  the  interpretation  of  the  next 
two  theorems.     The  analogues  of  these,  in  <p(x,  y},  do  not  hold. 

12-5     Zpx  +  IIty  =  U\f/y+2<px  =  2xHy(<px  +  $y)  =  2xlly(\l/y  +  <px) 

=  Uv2x(<px  +  \lry)  =  Uy^x(^y  +  <px). 

(1)  [4-3]  2<px  +  U\fsy  =  H.\f/y+2<px. 

(2)  By  proof  similar  to  (2)  in  12-2,  S^or  +  U\l/y  =  2xUy(<px  +  ty). 
And  by  proof  similar  to  (3)  in  12-2,  S<pz  +  U\f/y  =  SJIj,^  +  <px). 

(3)  By  proof  similar  to  (2)  in  12-4,  U\f/y+  2<px  =  Uy!,x(\f/y+  <f>x). 
And  by  proof  similar  to  (3)  in  12-4,  H\f/y  +  2<#r  =  Hv2x(<px  +  $y). 

"Either  for  some  x,  <px,  or  for  every  y,  \fsy"  is  equivalent  to  "For  some  x 
and  every  y,  either  <px  or  \{/y",  etc. 

12-6     I,<px 


(1)  [1-3] 

(2)  By  proof  similar  to  (2)  in  12-3,  2<px  xllty  =  2xny(<px  x^y}. 
And  by  proof  similar  to  (3)  in  12-3,  S<pz  xH\f/y  =  '2xlly(\l/y  x  <px). 

(3)  By  proof  similar  to  (2)  in  12-1,  Il^y  x2<px  =  n!/2x(^i/  x  <px). 
And  by  proof  similar  to  (3)  in  12-  1,  H\f/y  x?<px  =  HyZ^px  x  ^y}. 

"For  some  x,  <px,  and  for  every  y,  \[/y"  is  equivalent  to  "For  some  x  and 
every  y,  <px  and  \f/y",  etc. 
18 


258  A  Survey  of  Symbolic  Logic 

We  may  generalize  theorems  12-1-12-6  by  saying  that  for  functions 
of  the  type  ( <px  +  \l/y)  and  ( <px  x  \f/y)  the  order  of  operators  and  of  members 
in  the  function  is  indifferent;  and  for  propositions  of  the  type 

n 


the  operators  may  be  combined,  and  the  functions  combined  in  the  relation 
between  the  propositions. 

It  will  be  unnecessary  to  give  here  the  numerous  theorems  which  follow 
from  10-5-12-6  by  the  principles  pqcp,  pcp  +  q,  and  U<f>x  c2<px,  etc. 
For  example,  10-51, 

II  <px  x  Ufa  =  Ux(  <px  x  \f/x) 
gives  at  once 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

Etc.,  etc. 

And  12-2,  2<px  +  2\f/y  =  2x2y(<f>x  +  ty),  etc.,  gives' 
(1) 
(2) 
(3) 

(4)  TL+y  c 
(5) 
(6) 
Etc.,  etc. 

Another  large  group  of  theorems,  only  a  little  less  obvious,  follow  from 
the  combination  of  H<pxcH<px,  or  S#rcS#c,  with  H\!/y  c2ij/y,  giving 
by  5-3, 

(1) 

(2)  IL<pxx H\l/y  cll<pxx 

(3)  S  <pa;  +  Ityi/  c  S  ipx  + 

(4)  S^cxn^ycS^arx 
Etc.,  etc. 

Each  of  these  has  a  whole  set  of  derivatives  in  which  n  <px  +  H.\I/y  is  replaced 


Systems  Based  on  Material  Implication  259 

by  nxnv(^  +  ^2/),  etc.,  n«pa:xS^/  by  Ux2y((f>x  x^/),  etc.     We  give,  in 
summary  form,  the  derivatives  of  (2),  by  way  of  illustration: 

Any  one  of        c         any  one  of 
x  Hij/y  Ii(f>x  x  ?,\l/y 

x  ^y)         nxSj,(  <&  x  ^y) 
x  <px)         nxs,,(^  x  ^) 

x  pa)  Svn*(^  x  ^r) 

x  ^y)          synx(  <px  x  ^/) 

x  H<px 


n  <px  + 

nxll^(^  +  \f/y),  etc.,  etc. 

S  <px  +  Si/'?/ 

SS2V(^  +  ^i/),  etc.,  etc. 

n  .pa:  +  s^?/ 

Ux'2v(<(>x  +  $y),  etc.,  etc. 

This  table  summarizes  one  hundred  fifty-six  theorems,  and  these  are  only  a 
portion  of  those  to  be  got  by  such  procedures. 

Functions  of  the  type  of  (<pxx$y}  and  (<px  +  $y)  give  four  different 
kinds  of  implication  relation:  (1)  Hxlly(<px  c^/);  (2)  niSJ/(^a:  c^?/); 
(3)  2,xlly(<f>x  c\f/y)-,  and  (4)  SxS,,(^a;  c^y).  With  the  exception  of  the 
first,  these  relations  are  unfamiliar  as  "implications",  though  all  of  them 
could  be  illustrated  from  the  field  of  mathematics.  Nor  are  they  par- 
ticularly useful:  the  results  to  be  obtained  by  their  use  can  always  be  got 
by  means  of  material  implications  or  formal  implications.  Perhaps 
Uxny((px  c  \l/y]  is  of  sufficient  interest  for  us  to  give  its  elementary  properties. 

12-7    nzny(  <px  c  ^)  c  (  ex*  c 
[ii-oi]  nxny(^c 

x... 


...     [10-42] 
And  this  last  expression  is  equivalent  to  the  set 

<pxi  c  ll\f/y,   <px2  c  U\l/y,  <px3  c  U\{/y,  etc. 
12-71     {  nzn,,(  <pxcty)x  <pxn  }  c 
[9-4,  12-7] 


260  A  Survey  of  Symbolic  Logic 

If  for  every  x  and  every  y,  <?x  implies  ty,  and  for  some  given  x,  <px  is  true, 
then  $y  is  true  for  every  y. 

12-72     Uxllv(<pxc^y)  =  Z^rcltyi/  =  U-<f>x  +  H\l/y. 

(1)  If  UMvxcty),  then  [12-7]  <pxncUty. 
Hence  [5-991]  2<pxcUty. 
And  if  2<f>xcU\l/y,  then  [10-42]  Uv(2  <px  c  fy)  ,  and  hence  [10-41] 


(2)  [9-3] 

nznv(v*c  c^)  is  equivalent  to  "If  there  is  some  x  for  which   <po:  is  true, 
then  \f/y  is  true  for  every  y". 

12-73     (njI^zc^xIIjU^cfz)}  cIUIzOzcfz). 

[12-72]   If   njlj/^rc^?/)   and   Uyllz(^y  cfz),   then    S^ccH^y 
and  2^y  c  Hfiz. 
But    [10-21]    UtycZifsy.     Hence    [5-1]    S^cHfz,    and    [12-72] 


This  implication  relation  is  here  demonstrated  to  be  transitive.     In  fact, 
it  is,  so  to  speak,  more  than  transitive,  as  the  next  theorem  shows. 

12-74      {  (S  <px  c  Z^)  x  U»Uz(ty  c  fz)  }  c  n2nz(  pz  c  fz). 

[12-72]  n,,n,(^cfz)  =  Z^cHfz. 
And  [5-1]  if  Z^c  c  S^y  and  Z^y  cHfz,  then  Z^c  cHfz,  and  [12-72] 


12-75     {IUI,,(v*r 

[12-72]  nxnv 

And  [5-1]  if  Z^r  c  n^y  and  Ilty  c  n^z,  then  S^a:  c  Ilfz,  and  [12-72] 
nsnz(^a:  cfz). 

IV.    DERIVATION  OF  THE  LOGIC  OF  CLASSES  FROM  THE  CALCULUS  OF 
PROPOSITIONAL  FUNCTIONS 

The  logic  of  classes  and  the  logic  of  relations  can  both  be  derived  from 
the  logic  of  prepositional  functions.  In  the  present  chapter,  we  have 
begun  with  a  calculus  of  propositions,  the  Two-Valued  Algebra,  which 
includes  all  the  theorems  of  the  Boole-Schroder  Algebra,  giving  these 
theorems  the  prepositional  interpretation.  We  have  proved  that,  con- 
sidered as  belonging  to  the  calculus  of  propositions,  these  theorems  can 
validly  be  given  the  completely  symbolic  form:  "If  .  .  .  ,  then  ..." 


Systems  Based  on  Material  Implication  261 

being  replaced  by  ".  .  .  c  .  .  . ",  ".  .  .  is  equivalent  to  .  .  . "  by  ".  .  . 
=  .  .  . ",  etc.  The  Two-Valued  Algebra  does  not  presuppose  the  Boole- 
Schroder  Algebra;  it  simply  includes  it. 

Suppose,  then,  we  make  the  calculus  of  propositions — the  Two-Valued 
Algebra — our  fundamental  branch  of  symbolic  logic.  We  derive  from  it 
the  calculus  of  prepositional  functions  by  the  methods  of  the  last  two> 
sections.  We  may  then  further  derive  the  calculus  of  logical  classes,  and  a 
calculus  of  relations,  by  methods  which  are  to  be  outlined  in  this  section 
and  the  next. 

The  present  section  will  not  develop  the  logic  of  classes,  but  will  present 
the  method  of  this  development,  and  prove  the  possibility  and  adequacy 
of  it.  At  the  same  time,  certain  differences  will  be  pointed  out  between 
the  calculus  of  classes  as  derived  from  that  of  prepositional  functions  and 
the  Boole-Schroder  Algebra  considered  as  a  logic  of  classes.  In  order  to 
distinguish  class-symbols  from  the  variables,  x,  y,  z,  in  prepositional  func- 
tions, we  shall  here  represent  classes  by  a,  /3,  7,  etc. 

For  the  derivation  of  the  logic  of  classes  from  that  of  propositional 
functions,  a  given  class  is  conceived  as  the  aggregate  of  individuals  for 
which  some  propositional  function  is  true.  If  <pxn  represent  "x  is  a  man", 
then  the  aggregate  of  x's  for  which  <px  is  true  will  constitute  the  class  of 
men.  If,  then,  z(<pz)  represent  the  aggregate  of  individuals  for  which  the 
propositional  function  <pz  is  true,  z(<pz)  will  be  "the  class  determined  by 
the  function  <pz",  or  "the  class  determined  by  the  possession  of  the  char- 
acter <p".ls  We  can  use  a,  j8,  7,  as  an  abbreviation  for  z(<pz),  z(\l/z),  z(£z), 
etc.  a  =  z(<pz)  will  mean  "a  is  the  class  determined  by  the  function  <pz". 
(In  this  connection,  we  should  remember  that  <px  and  <pz  are  the  same 
function.) 

The  relation  of  an  individual  member  of  a  class  to  the  class  itself  will 
be  symbolized  by  c.  xn  e  a  represents  "xn  is  a  member  of  a" — or  briefly 
"xn  is  an  a".  This  relation  can  be  defined. 

13  We  here  borrow  the  notation  of  Principia.  The  corresponding  notation  of  Peirce 
and  Schroder  involves  the  use  of  S,  which  is  most  confusing,  because  this  S  has  a  meaning 
entirely  different  from  the  S  which  is  an  operator  of  a  propositional  function.  But  in 
Principia,  z(<pz)  does  not  represent  an  aggregate  of  individuals;  it  represents  "z  such 
that  <f>z".  And  z(<pz)  is  not  a  primitive  idea  but  a  notation  supported  by  an  elaborate 
theory.  Our  procedure  above  is  inelegant  and  theoretically  objectionable:  we  adopt  it 
because  our  purpose  here  is  expository  only,  and  the  working  out  of  an  elaborate  technique 
would  impede  the  exposition  and  very  likely  confuse  the  reader.  As  a  fact,  a  more  satis- 
factory theory  on  this  point  makes  no  important  difference. 


262  A  Survey  of  Symbolic  Logic 


13-01     xni&(<pz)  =  <pxn        Def. 

"xn  is  a  member  of  the  class  determined  by  <pz"  is  equivalent  to  "  <pxn  is 
true". 

(For  convenience  of  reference,  we  continue  to  give  each  definition  and 
theorem  a  number.) 

The  relation  "a  is  contained  in  0"  is' the  relation  of  the  class  a  to  the 
class  0  when  every  member  of  a  is  a  member  of  0  also.  We  shall  symbolize 
"a  is  contained  in  0 "  by  a  c  0.  The  sign  c  between  a  and  0,  or  between 
z(<pz)  and  2(^2),  will  be  "is  contained  in";  c  between  propositions  will 
be  "implies  ",  as  before.  xn  e  a  is,  of  course,  a  proposition;  x  e  a,  a  propo- 
sitional function. 

13-02     a  c  0  =  Ilx(x  €  a  c  x  £  0)         Def. 

a  c  0  is  equivalent  to  "For  every  x,  'a:  is  an  a'  implies  'x  is  a  0'". 

Hx(x  e  a  c  x  e  0)  is  a  formal  implication.  It  will  appear,  as  we  proceed 
that  the  logic  of  classes  is  the  logic  of  the  formal  implications  and  formal 
equivalences  which  obtain  between  the  propositional  functions  which  deter- 
mine the  classes. 

13-03      (a  =  j8)  =  Ux(x  e  a  =  x  e  0)          Def. 

a  =  0  is  equivalent  to  "For  every  x,  'x  is  a  member  of  a'  is  equivalent 
to  'x  is  a  member  of  /3'".  a  =  0  thus  represents  the  fact  that  a  and  0 
have  the  same  extension — i.  e.,  consist  of  identical  members. 

xn  e  a,  a  c  0,  and  a  =  /3  are  assertable  relations — propositions.  But 
the  logical  product  of  two  classes,  and  the  logical  sum,  are  not  assertable 
relations.  They  are,  consequently,  defined  not  by  means  of  propositions 
but  by  means  of  functions. 

.13-04     ax/3  =  £{(x€ct)  x  (x  e  0)}          Def. 

The  product  of  two  classes,  a  and  0,  is  the  class  of  x's  determined  by  the 
propositional  function  "x  is  an  a  and  x  is  a  0".  The  class  of  the  x's  for 
which  this  is  true  constitute  a  x  0,  the  class  of  those  things  which  are 
both  a's  and  0's. 

The  relation  x  between  a  and  0  is,  of  course,  a  different  relation  from  x 
between  propositions  or  between  propositional  functions.  A  similar  remark 
applies  to  the  use  of  + ,  which  will  represent  the  logical  sum  of  two  classes, 
as  well  as  of  two  propositions  or  propositional  functions.  This  double 
use  of  symbols  will  cause  no  confusion  if  it  be  remembered  that  a  and  0, 
z(<pz~)  and  2(^-2),  etc.,  are  classes,  while  a:  e  a  is  a  propositional  function, 
and  xn  e  a,  a  c  0,  and  a  =.  0  are  propositions. 


Systems  Based  on  Material  Implication  263 

13-05     a  +  ft  =  x{(x  e  a)  +  (x  e  ft)}         Def. 

The  sum  of  two  classes,  a  and  ft,  is  the  class  of  x's  such  that  at  least  one 
of  the  two,  (x  is  an  a'  and  'x  is  a  ft',  is  true,  or  loosely,  the  class  of  x's 
such  that  either  x  is  an  a  or  x  is  a  ft. 

The  negative  of  a  class  can  be  similarly  defined: 

13-06     -a  =  x-(xea)         Def. 

The  negative  of  a  is  the  class  of  x's  for  which  'x  is  an  «'  is  false. 

The  "universe  of  discourse",  1,  may  be  defined  by  the  device  of  selecting 
some  propositional  function  which  is  true  for  all  values  of  the  variable. 
Such  a  function  is  (£x  cfaj),  whatever  propositional  function  $x  may  be. 

13-07     1  =  x($xc$x)         Def. 

1  is  the  class  of  x's  for  which  £x  implies  £x.u    Since  this  is  always  true,  1  is 

the  class  of  all  x's.     The  "null-class",  0,  will  be  the  negative  of  1. 

13-08     0  =  -1         Def. 

That  is,  by  13-06,  0  =  x  -(£x  c  £x),  and  since  -(fzcfz)  is  false  for  all 

values  of  x,  the  class  of  such  x's  will  be  a  class  with  no  members. 

Suppose  that  a  =  z(<pz)  and  ft  =  z($z).  Then,  by  13-01,  xn  e  a  =  <pxn. 
Hence  a  c  ft  will  be  Hx(<p%  c  \f/x),  and  a  =  ft  will  be  Iix(<px  —  \f/x).  This 
establishes  at  once  the  connection  between  the  assertable  relations  of 
classes  and  formal  implication  and  equivalence.  To  illustrate  the  way  in 
which  this  connection  enables  us  to  derive  the  logic  of  classes  from  that  of 
propositional  functions,  we  shall  prove  a  number  of  typical  theorems. 

It  will  be  convenient  to  assume  for  the  whole  set  of  theorems: 

a  =  z(<pz),         ft  =  z($z),         7  =  $(&) 
13-1     0  =  &-($xc$x). 

0  = -1.     Hence  [13-06]  0  =  x -(x  el). 
[13  •  01  •  07]  x  e  1  =  £  x  c  £x.     Hence  0  =  x  -(fa;  c  fc) . 

13-2     Ux(xel). 

[13-01-06]    Xntl    =    (frnCfrJ. 

Hence  Ilx(x  e  1)  =  Ux(£x  c£x). 

But  [2-2]  £xn  c£xn.     Hence  [10-23]  TLx($x  cfa:). 

Every  individual  thing  is  a  member  of  the  "universe  of  discourse". 

14  This  defines,  not  the  universe  of  discourse,  but  "universe  of  discourse", — the  range 
of  significance  of  the  chosen  function,  f .  With  1  so  denned,  propositions  which  involve 
the  classes  £(<t>z),  z(\j/z),  etc.,  and  1,  will  be  significant  whenever  <p,  \l/,  etc.,  and  f  have 
the  same  range,  and  true  if  significant. 


264  A  Survey  of  Symbolic  Logic 

13-3     nx-(zeO). 

[1$-01  -06  -07]  zncO  =  -(r**cfa:»). 
Hence  [3-2]  -(zneO)  =  (r*»cfa:B). 
But  fzn  c  fa;n.     Hence  [10  •  23]  Ux(fx  c  fx),  and  Hz  -(z  e  0). 

For  every  x,  it  is  false  that  x  c  0  —  no  individual  is  a  member  of  the  null- 
class. 

13-4     «cl. 

[13-01-06]  xnel  =  ({XnCtxJ. 
Since  a  =  z(<pz),  [13-01]  xn  e  a  =  <pxn. 

[9-33]    (ranCfSn)   C[^nC(fZnCrSw)]. 

Hence  since  £rn  c  £rn,  ^n  c  (£xn  c  fzn). 

Hence  [10-23]  nx[^r  c(fx  cfa)],  and  [13-2]  ad. 

Any  class,  a,  is  contained  in  the  universe  of  discourse.  It  will  be  noted 
(13-2  and  13-4)  that  individuals  are  members  of  1,  classes  are  contained  in  1. 
In  the  proof  of  13-4,  we  make  use  of  9-33,  "A  true  proposition  is  implied 
by  any  proposition  ".  £xn  c  £xn  is  true.  Hence  it  is  implied  by  <pxn.  And 
since  this  holds,  whatever  value  of  x,  xn  may  be,  therefore, 


But  <px  is  the  function  which  determines  the  class  a;  £x  c£x,  the  function 
which  determines  1.  Hence  <pxn  is  xn  t  a,  and  £xn  c£xn  is  xn  c  1.  Conse- 
quently we  have  IIX  (x  e  a  ex  el).  And  by  the  definition  of  the  relation 
"is  contained  in",  this  is  a  c  1. 

13-5     Oca. 

[9  •  1]  -(#„  e  0)  is  equivalent  to  (xn  e  0)  =0. 
Hence  [13-3]  (xn  c  0)  =0,  and  [9-32]  (xn  c  0)  c  <pxn. 
Hence  [13-01]  xn  c  0  c.rn  c  a,  and  [10-23]  Ux(x  t  0  ex  e  a). 
Hence  [13-02]  0  c  a. 

The  null-class  is  contained  in  every  class,  a.  In  this  proof,  we  use  9-32, 
"A  false  proposition  implies  any  proposition".  -(f.rncfa:n)  is  false,  and 
hence  implies  <p.rn.  But  ~(fxc£x)  is  the  function  which  determines  0; 
and  <px,  the  function  which  determines  a.  Hence  0  c  a. 

The  proofs  of  the  five  theorems  just  given  are  fairly  typical  of  those 
which  involve  0  and  1.  But  the  great  body  of  propositions  make  more 
direct  use  of  the  connection  between  the  relations  of  classes  and  formal 
implications  or  equivalences.  This  connection  may  be  illustrated  by  the 
following: 


Systems  Based  on  Material  Implication  265 

13-6     z(<pz)  c  2(^2)  =  Hx(<f>x  c^ar). 

[13-02]  2(^2)  cz(^z)  =  IIx[ar  e  2(>2)  ex  e  2(^2)]. 
[13-01]  arn  e  z(<pz)  =  <pxn,  and  arn  e  2(1^2)  =  ^arn. 
Hence  [2-1]  Hx[x  e  2(^2)  car  c  2(^2)]  =  Hx(<px  c  ^ar). 

"The  class  determined  by  #s  is  contained  in  the  class  determined  by  ^2" 
is  equivalent  to  "For  every  x,  <f>x  implies  \f/x". 

[13-03]  [2(>2)  =  2(^2)]  =  IIx[ar  e  z(<pz)  =  x  e  2(^2)]. 
'[lo"Ulj  arra  €  2^^>2j  ==  <pXn}  and  xn  c  z\y/z)  ==  \f/xn* 
Hence  Ux[x  e  2(^2)  =  x  e  2(^2)]  =  IIx(^a:  =  ^x). 

'   "The  class  determined  by  tpz  is  equivalent  to  the  class  determined  by  ^2" 
is  equivalent  to  "For  every  x,  <px  is  equivalent  to  \j/x". 

13-8     (ac  |8)  =  (-|8  c-a). 

[10-66]  Hx[a:  e  2(^2)  car  e  2(^2)]   =  Ux{-[x  e  2(^2)]  c-[ar  e  2(^2)]}. 
Hence  nx(a;  e  a  ex  e  |8)  =  Hx[-(x  e  |8)  c-(ar  c  a)]. 
[13-01-06]  -(x  e  a)  =  xe  -a,  and  -(ar  e  0)  =  z  e  -j8. 
Hence  [13-02]  («c/3)  =  (-/Sc-a). 

[13-6]  (a  c  j3)  =  nx(^>a:  c^x),  (/3  c  7)  =  nx(^a:  c  far),  and  (a  c  7) 
=  nx(<p£  c  far). 
And  [10-65]  [IIx(^ar  c  ^ar)  x  nx(i^ar  c  far)]  c  nx(^>ar  c  far). 

The  relation  "is  contained  in"  is  transitive.     13-9  is  the  first  form  of  the 
syllogism  in  Barbara.     The  second  form  is: 

13-91     [(a  c  8)  x  (arn  e  a)]  c  (ar    e  fi). 

[13-6]  (aCjS)  =  nx(<par  c^ar). 
[13-01]  (xnca)  =  <pxn,  and  (xn  e  /3)  =  ^arre. 
And  [10-611]  [nx(^ar  c^ar)  x  ^arn]  c^arn. 

If  the  class  a  is  contained  in  the  class  /3,  and  a:re  is  a  member  of  a,  then  arn 
is  a  member  of  /3. 

13-92     [(a  =  /3)  x  (/3  =  7)]  c  (a  =  7). 

(a  =  7)  =  nx(^ar  =  far). 

And  [10-68]  [nx(<par  =  ^ar)  xnx(^ar  =  far)]  cnx(^>x  =  far). 

The  last  three  theorems  illustrate  particularly  well  the  direct  connection 


266  A  Survey  of  Symbolic  Logic 

between  formal  implications  and  the  relations  of  classes.  13-6  and  13-7 
are  alternative  definitions  of  a  c  /3  and  a  =  /3.  Similar  alternative  defini- 
tions of  the  other  relations  would  be  :  16 


We  may  give  one  theorem*  especially  to  exemplify  the  way  in  which 
every  proposition  of  the  Two-  Valued  Algebra,  since  it  gives,  by  10-23,  a 
formal  implication  or  equivalence,  gives  a  corresponding  proposition  con- 
cerning classes.  We  choose  for  this  example  the  Law  of  Absorption. 

13-92     [a+(aX/3)]  =  a. 

[13-04-05]  [a+(aXj8)]  =  x{(xea)  +  [(xta)  x(.re/3)]}. 
Hence  [13-01]  [xn  e  [a+  (a  x/3)]} 

=  {  (*„  e  a)  +  [(z«  e  a)  x(arnej8)]}.     (1) 
But  [13-03]  {[a+(ax|8)]  =  a} 

=  IIxC{(a;e  a)  +  [(«««)  x(a?e/S)])  =  (zea)].     (2) 
But  [13-01]  (xn  e  a)  =  <pzn,  (a;B  e  0)  =  fe,  and  by  (2), 

{[a+  (a  Xj8)]  =  a}  =  na{[$sr  +  (<px  x 
But  [5-4]  [$«»  +  (0tf»x#eB)] 
Hence  [10-23]  Ux{[<px  +  (<?x 


All  but  the  last  two  lines  of  this  proof  are  concerned  with  establishing 
the  connection  between  [a  +  (a  x/3)]  =  «  and  the  formal  equivalence 


Once  this  connection  is  made,  we  take  that  theorem  of  the  Two-Valued 
Algebra  which  corresponds  to  [a  +  (a  x  /3)]  =  a,  namely  5-4,  (p  +  p  q)  =  p, 
substitute  in  it  <pxn  for  p  and  \f/xn  for  q,  and  then  generalize,  by  10-23,  to 
the  formal  equivalence  which  gives  the  proof.  An  exactly  similar  pro- 
cedure will  give,  for  most  theorems  of  the  Two-  Valued  Algebra,  a  corre- 
sponding theorem  of  the  calculus  of  classes.  The  exceptions  are  such 
propositions  as  p  =  (p  =  1),  which  unite  an  element  p  with  an  implication 
or  an  equivalence.  In  other  words,  every  theorem  concerning  classes  can 
be  derived  from  its  analogue  in  the  Two-  Valued  Algebra. 

We  may  conclude  our  discussion  of  the  derivation  of  the  logic  of  classes 

15  As  a  fact,  these  definitions  would  be  much  more  convenient  for  us,  but  we  have 
chosen  to  give  them  in  a  form  exactly  analogous  to  the  corresponding  definitions  of  Prin~ 
cipia  (see  i,  p.  217). 


Systems  Based  on  Material  Implication  267 

from  the  logic  of  prepositional  functions  by  deriving  the  set  of  postulates 
for  the  Boole-Schroder  Algebra  given  in  Chapter  II.  This  will  prove  that, 
beginning  with  the  Two-  Valued  Algebra,  as  a  calculus  of  propositions,  the 
calculus  of  classes  may  be  derived.  This  procedure  may  have  the  appear- 
ance of  circularity,  since  in  Section  I  of  this  chapter  we  presumed  the 
propositions  of  the  Boole-Schroder  Algebra  without  repeating  them.  But 
the  circularity  is  apparent  only,  since  the  Two-Valued  Algebra  is  a  distinct 
system. 

The  postulates  of  Chapter  II,  in  a  form  consonant  with  our  present 
notation,  can  be  proved  so  far  as  these  postulates  express  symbolic  laws. 
The  postulates  of  the  existence,  in  the  system,  of  -a  when  a.  exists,  of  a  x  /3 
when  a  and  /3  exist,  and  of  the  class  0,  must  be  supposed  satisfied  by  the 
fact  that  we  have  exhibited,  in  their  definitions,  the  logical  functions  which 
determine  a  x  /3,  -a,  and  O.16 

14-2     (a  xa)  =  a. 

[13-01]  xn  e  a  =  <pxn. 

Hence  [13-04]  xn  e  (a  x  a)  =  [(xn  e  a)  x  (xn  e  a)]  =  (<pxn  x  <pxn}. 
Hence  [13-03]  [(a  Xct)  =  a]  =  Hx{[x  e  (a  x  a)]  =  x  e  a} 

=  nx[(<f>x  x  <px)  =  <px]. 

But  [1-2]  (<pxn  x  tpxn)  =  <pxn. 
Hence  [10-23]  Ux[(<px  x  <px)  =  <px]. 

14-3      (aX/3)  =  (0x«). 

[13-03]  [(ax/3)  =  (/3  x  a)]  =  Ux{(x  t  (a  x  0)]  =  [se(/3xa)]} 
=  n*{[(zea)  x(aj€j8)]  =  [(x  e  /3)  x(sea)]}. 

[13-01-04] 

Hence  [13-01]  [(ax0)  =  08  x  a)]  =  ns[(#rx#c)  =  (#e  x  <px)]. 
But  [1-3]  (<pxnx\l/Xn)  =  (txnx<pXn). 
Hence  [10-23]  ttx[(<px  x^z)  =  (#»  x  <px)]. 

14-4     (aX|8)  XT  =  a  x  (,3x7). 

[13-03]  [(ax/3)  x7  =  ax(/3x7)]  = 
=  n.CJOJeKaX^XT]}   =   (ze  [ax  03X7)]}] 


[13-01-04] 
Hence  [13-01]  [(ax/3)  x7  =  a  x  (18x7)] 

=  Hx{[(<f>x  x  #r)  x  ^]  =  [^.r  x  (^ 


16  A  more  satisfactory  derivation  of  these  existence  postulates  is  possible  when  the 
theory  of  prepositional  functions  is  treated  in  greater  detail.     See  Prindpia,  I,  pp.  217-18. 


268  A  Survey  of  Symbolic  Logic 

But  [1-04]  (<pxn  x  fan)  x  &rn  =  <pxn  x  (fan  x  &rn). 
Hence  [10-23]  n*{[(?o:  x#r)  xfcr]  =  [#r  x(#e  x{a:)]}. 

14-5     axO  =  0. 

[13-1-01]  xntO  =  -(fo-nCfav). 
[13-03-04]  [axO  =  0]  =  nx{[(a:nea)x(.rneO)]  =  (zneO)} 

=  II.{[*ax-(fa:cfaO]  = -(fa;  c far) } .  [13-01] 
But  [2-2,  9-01]  (f.rncfzn)  =  1,  and  [3-2]  -(fzncfzn)  =  0. 
Hence  [1-5]  [<pxn  x-(fzB  cfa;n)]  =  0  =  -(fajB  cfa:»). 
Hence  [10-23]  nx{[^  x-(fz  cfar)]  =  ~(fxcfx)}. 

0,  in  the  fourth  and  fifth  lines  of  the  above  proof,  is  the  0  of  the  Two- Valued 
Algebra,  not  the  0  of  the  calculus  of  classes.  Since  the  general  method  of 
these  proofs  will  now  be  clear,  the  remaining  demonstrations  can  be  some- 
what abbreviated. 

14-61      [(aX-0)  =  0]c[(ax0)  =  a]. 

[13 -01 -02 -04 -06]  The  theorem  is  equivalent  to 

n*{[(#e  x-^ar)  =  (a:  e  0)]  c  [(<px  x^c)  =  <px]} 
But  [13-3]  Ux  -(x  e  0),  and  hence  [9-1]  Ux[(x  €  0)  =  0]. 
Hence  the  theorem  is  equivalent  to 

nx{[(#r  x-#c)  =  (x  eQ)]c[(<pxx\l/x')  =  <px]} 
But  [13-3]  na-(areO),  and  hence  [9-1]  Ux[(xeO)  =  0]. 
Hence  the  theorem  is  equivalent  to 

TLx{[(<px  x-\frx)  =  0]c[(v?a:x^)  =  <px]} 
But  [1-61]  [(<pxn  x-^n)  =  0]  c[(^o;n  x^xn)  =  <pxn]. 
HenceJlO-23]  Q.E.D. 

14-62      {[(ax/8)  =  a]  x[(«x-/3)  =  a]}  c  (a  =  0). 
The  theorem  is  equivalent  to 

n*[{[(#cx#c)  =  ^]x[(^x-^)  =  <px]}  c[<px  =  (a?«0)]J 
But  [13-3,  9-1]  lUfreO)  =  0]. 
Hence  the  theorem  is  equivalent  to, 

nx[{[(v5#  x^ar)  =  <p£]  x[(^?a:  x-^ar)  =  ^x]}  c(<px  =  0)] 
But  [1-62]  {[(*«»  X#CB)  =  ^n]  x[(^cn  x-^.rn)  =  ^ar«]}  c(^n  =  0). 
Hence  [10-23]  Q.E.D. 

The  definition,  1  =  -0,  follows  readily  from  the  definition  given  of  0  in 
this  section.     The  other  two  definitions  of  Chapter  II  are  derived  as  follows: 

14-8     (a+/3)  =  -(-ax-j8). 

The  theorem  is  equivalent  to  Ux[(<px  +  fa)  =  -(-<px  x-^ar)]. 


Systems  Based  on  Material  Implication  269 

But  [1-8]  (<pxn  +  txn)  =  -(-^nx-^zn). 
Hence  [10-23]  Q.E.D. 

14-9     (aC(8)  =  [(ax/3)  -  a]. 

The  theorem  is  equivalent  to  Ux(<px  c\l/x)  =  Hz[(<px  x\l/x)  =  <px]. 
But  [1-9]  (<f>xncipXn)  is  equivalent  to  [(<pxn  x\f/Xn)  =  <pxn]. 
Hence  [10-23-69]  Q.E.D. 

Since  the  postulates  and  definitions  of  the  calculus  of  classes  can  be 
deduced  from  the  theorems  of  the  calculus  of  prepositional  functions,  it 
follows  that  the  whole  system  of  the  logic  of  classes  can  be  so  deduced. 
The  important  differences  between  the  calculus  of  classes  so  derived  and 
the  Boole-Schroder  Algebra,  as  a  logic  of  classes,  are  two:  (1)  The  Boole- 
Schroder  Algebra  lacks  the  e-relation,  and  is  thus  defective  in  application, 
since  it  cannot  distinguish  the  relation  of  an  individual  to  the  class  of 
which  it  is  a  member  from  the  relation  of  two  classes  one  of  which  is  con- 
tained in  the  other;  (2)  The  theorems  of  the  Boole-Schroder  Algebra 
cannot  validly  be  given  the  completely  symbolic  form,  while  those  of  the 
calculus  of  classes  derived  from  the  calculus  of  prepositional  functions  can 
be  given  this  form.17  « 

V.    THE  LOGIC  OF  RELATIONS 

The  logic  of  relations  is  derived  from  the  theory  of  propositional  func- 
tions of  two  or  more  variables,  just  as  the  logic  of  classes  may  be  based 
upon  the  theory  of  propositional  functions  of  one  variable. 

A  relation,  R,  is  determined  in  extension  when  we  logically  exhibit  the 
class  of  all  the  couples  (x,  y)  such  that  x  has  the  relation  R  to  y.  If  <p(x,  y) 
represent  "x  is  parent  of  ?/",  then  x  y[<p(x,  y)]  is  the  relation  "parent  of". 
This  defines  the  relation  in  extension:  just  as  the  extension  of  "red"  is 
the  class  of  all  those  things  which  have  the  property  of  being  red,  so  the 
extension  of  the  relation  "parent  of"  is  the  class  of  all  the  parent-child 
couples  in  the  universe.  A  relation  is  a  property  that  is  common  to  all  the 
couples  (or  triads,  etc.)  of  a  certain  class;  the  extension  of  the  relation  is, 
thus,  the  class  of  couples  itself.  The  calculus  of  relations,  like  the  calculus 
of  propositions,  and  of  classes,  is  a  calculus  of  extensions. 

-17  Oftentimes,  as  in  Schroder,  Alg.  Log.,  I,  the  relations  of  propositions  in  the  algebra 
of  classes  have  been  represented  in  the  symbols  of  the  propositional  calculus  before  that 
calculus  has  been  treated  otherwise  than  as  an  interpretation  of  the  Boole-Schroder  Algebra. 
But  in  such  a  case,  if  these  symbols  are  regarded  as  belonging  to  the  system,  the  procedure 
is  invalid. 


270  A  Survey  of  Symbolic  Logic 

We  assume,  then,  the  idea  of  relation :  the  relation  R  meaning  the  class 
of  couples  (x,  y)  such  that  x  has  the  relation  R  to  y. 

R  =  A  y(x  R  y),        S  =  w  z(w  S  2),        etc. 

This  notation  is  simpler  and  more  suggestive  than  R  =  A  y[<p(x,  y)], 
S  =  w  z[^(w,  2)],  but  it  means  exactly  the  same  thing.  A  triadic  relation, 
T,  will  be  such  that 

T  =  xyz[T(x,y,z}} 

or  T  is  the  class  of  triads  (x,  y,  z)  for  which  the  prepositional  function 
T(x,  y,  2)  is  true.  But  all  relations  can  be  defined  as  dyadic  relations.  A 
triadic  relation  can  be  interpreted  as  a  relation  of  a  dyad  to  an  individual — 
that  is  to  say,  any  function  of  three  variables,  T(x,  y,  2),  can  be  treated  as  a 
function  of  two  variables,  the  couple  (x,  y)  and  2,  or  x  and  the  couple  (y,  2) . 
This  follows  from  the  considerations  presented  in  concluding  discussion  of 
the  theorems  numbered  11-,  in  section  III.18  Similarly,  a  tetradic  relation 
can  be  treated  as  a  dyadic  relation  of  dyads,  and  so  on.  Hence  the  theory 
of  dyadic  relations  is  a  perfectly  general  theory. 

Definitions  exactly  analogous  to  those  for  classes  can  be  given. 

15  -01     (x,  y}n  e  2  w[R(z,  w)]  =  R(x,  y)n.        Def. 

It  is  exactly  at  this  point  that  our  theoretical  considerations  of  the  equiva- 
lence of  <f>(x,  y)n  and  v(xr  y,}  becomes  important.  For  this  allows  us  to 
treat  R(x,  y),  or  (xRy),  as  a  function  of  one  or  of  two  variables,  at  will; 
and  by  11-07,  we  can  give  our  definition  the  alternative  form: 

15  -01     (xm  yn}tzw(zRw)  =  xmR  yn.        Def. 

"The  couple  (xm  yn)  belongs  to  the  field,  or  extension,  of  the  relation  deter- 
mined by  (2  R  w) "  means  that  xm  R  yn  is  true. 

15-02    RcS  =  Ilx,  y[(x  R  y)  c  (x  S  y)].        Def. 
This  definition  is  strictly  parallel  to  13-02, 

(a  c  /3)  =  Tlx  (x  e  a  C  x  e  /3) 

because,  by  15-01,  (x  R  y}  is  (x,  y)  e  R  and  (x  S  y)  is  (x,  y)  c  S.  A  similar 
remark  applies  to  the  remaining  definitions. 

15-03     (R  =  S)  =  Hx,  v[(x  Ry)  =  (xS  y}].        Def. 
R  and  S  are  equivalent  in  extension  when,  for  every  x  and  every  y,  (x  R  y) 
and  (x  S  y)  are  equivalent  assertions. 
18  See  above,  pp.  253  ff. 


Systems  Based  on  Material  Implication  271 

15-04     RxS  =  xy[(xRy)  x(xSy)].        Def. 

The  logical  product  of  two  relations,  R  and  S,  is  the  class  of  couples  (x,  y. 
such  that  x  has  the  relation  R  to  y  and  x  has  the  relation  S  to  y.  If  R  is 
"friend  of",  and  S  is  "colleague  of",  R  x  S  will  be  "friend  and  colleague  of") 

15-05    R  +  S  =  xy[(xRy)  +  (xSy)].        Def. 

The  logical  sum  of  two  relations,  R  and  S,  is  the  class  of  couples  (x,  y)  such 
that  either  x  has  the  relation  R  to  y  or  x  has  the  relation  S  to  y.  R  +  S 
will  be  " Either  R  of  or  S  of". 

15-06    -R  =  xy-(xRy}.        Def. 

-.R  is  the  relation  of  x  to  i/  when  a:  does  not  have  the  relation  R  to  y. 

It  is  important  to  note  that  R  x  S,  R  +  S,  and  -R  are  relations :  x(R  x  S)y, 
x(R+S)y,  and  x-Ry  are  significant  assertions. 

The  "universal-relation"  and  the  "null-relation"  are  also  definable 
after  the  analogy  to  classes. 

15-07     1  =  xy[t;(x,y)ct(x,y)}.        Def. 

x  has  the  universal-relation  to  y  in  case  there  is  a  function,  f,  such  that 

£•  (x,  y}  c£(x,  y},  i.  e.,  in  case  x  and  y  have  any  relation. 

15-08     0  =  -1.         Def. 

Of  course,  0,  1,   +  and   x  have  different  meanings  for  relations  from  their 
meanings  for  classes  or  for  propositions.     But  these  different  meanings  o 
0,  +  ,  etc.,  are  strictly  analogous. 

As  was  pointed  out  in  Section  III  of  this  chapter,  for  every  theorem 
involving  functions  of  one  variable,  there  is  a  similar  theorem  involving 
functions  of  two  variables,  due  to  the  fact  that  a  function  <p(x,  y)  may  be 
regarded  as  a  function  of  the  single  variable  (x,  y).  Consequently,  for 
each  theorem  of  the  calculus  of  classes,  there  is  an  exactly  corresponding 
theorem  in  the  calculus  of  relations.  We  may,  then,  cite  as  illustrations  of 
this  calculus  the  analogues  of  the  theorems  demonstrated  to  hold  for 
classes;  and  no  proofs  will  here  be  necessary.  These  proofs  follow  from  the 
theorems  of  Section  III,  numbered  11-,  exactly  as  the  proofs  for  classes 
are  given  by  the  corresponding  theorems  in  Section  II,  numbered  10  • . 

15-1     0  =  a0-[f(a?,y)cr(a:,y)]. 

The  null-relation  is  the  relation  of  x  to  y  when  it  is  false  that  f  (x,  y)  implies 
f  (x,  y),  i.  e.,  when  x  has  no  relation  to  y.19  Of  course,  there  is  no  such 
(x,  y)  couple  which  can  significantly  be  called  a  couple. 

19  As  in  the  case  of  the  1  and  0  of  the  class  calculus,  the  1  and  0  of  relations,  defined  as 


272  A  Survey  of  Symbolic  Logic 

15-2     nx,  ,[(*,  y)  €  1]. 

Every  couple  is  a  member  of  the  universe  of  couples,  or  has  the  universal 

(dyadic)  relation. 

15-3     n,,  „  -[(*,  y)  e  0]. 
No  couple  has  the  null-relation. 
15-4     Rcl. 
15-5    Ocfl. 

Every  relation,  R,  is  implied  by  the  null-relation  and  implies  the  universal 
relation;  or,  whatever  couple  (x,  y)  has  the  null-relation  has  also  the 
relation  R,  and  whatever  couple  has  any  relation,  R,  has  also  the  universal- 
relation. 

15-6     (flcS)  =  Hx,y[(xRy-)c(xSy-)]. 

For  relations,  R  cS  is  more  naturally  read  "  R  implies  S"  than  "  R  is  con- 
tained in  S".  By  15-6,  "R  implies  S"  means  "For  every  x  and  every  y, 
if  x  has  the  relation  R  to  y,  then  x  has  the  relation  S  to  y".  Or  " R  implies 
S"  means  "Every  (x,  y)  couple  related  by  R  are  also  related  by  S". 

15-7     (R  =  S)  =  nx,  y[(x  Ry}  =  (x  S  y)}. 

Two  relations,  R  and  S,  are  equivalent  when  the  couples  related  by  R  are 
also  related  by  S,  and  vice  versa  (remembering  that  =  is  always  a  reciprocal 
relation  c). 

15-8     (RcS)  =  (-Sc-fl). 

If  the  relation  R  implies  the  relation  S,  then  when  S  is  absent  R  also  will 

be  absent. 

15-9     [(R  cS)x(Sc  T)]  c(Rc  T). 

The  implication  of  one  relation  by  another  is  a  transitive  relation. 

15-91     [(R  c  S)  x  (xm  R  yn}}  c  (arw  8  y.n). 

If  .R  implies  /S  and  a  given  couple  are  related  by  R,  then  this  couple  are 

related  also  by  S. 

15-92     [(R  =  S)  x  (S  =  T)]  c(R=  T). 
The  equivalence  of  relations  is  transitive. 

If  it  be  supposed  that  the  postulates  concerning  the  existence  of  rela- 
tions are  satisfied  by  exhibiting  the  functions  which  determine  them,  then, 

we  have  defined  them,  are  such  that  propositions  involving  them  are  true  whenever  sig- 
nificant, and  significant  whenever  the  prepositional  functions  determining  the  functions  in 
question  have  the  same  range. 


Systems  Based  on  Material  Implication  273 

as  in  the  case  of  classes,  we  can  derive  the  postulates  (or  remaining  postu- 
lates) for  a  calculus  of  relations  from  the  theorems  of  the  calculus  of  prepo- 
sitional functions.  The  demonstrations  would  be  simply  the  analogues  of 
those  already  given  for  classes,  and  may  be  omitted, 

16-2     (RxR)  =  R. 

16-3     (RxS)  =  (SxK). 

16-4     (RxS)xT  =  Rx(SxT). 

16-5     RxQ  =  0. 

16-61     [(Rx-S)  =  0]c[(JRxS)  =  R]. 

16-62     {[(flxS)  =  R]  x[(Rx-S)  =  R]}  c  (fl  =  0). 

16-8     (R  +  S)  =  -(-Rx-S). 

16-9     (RcS)  =  [(RxS)  =  R]. 

These  theorems  may  also  be  taken  as  confirmation  of  the  fact  that 
the  Boole-Schroder  Algebra  holds  for  relations.  In  fact,  "calculus  of 
relations"  most  frequently  means  just  that — the  Boole-Schroder  Algebra 
with  the  elements,  a,  b,  c,  etc.,  interpreted  as  relations  taken  in  extension. 

So  far,  the  logic  of  relations  is  a  simple  analogue  of  the  logic  of  classes. 
But  there  are  many  properties  of  relations  for  which  classes  present  no 
analogies,  and  these  peculiar  properties  are  most  important.  In  fact, 
the  logistic  development  of  mathematics,  worked  out  by  Peirce,  Schroder, 
Frege,  Peano  and  his  collaborators,  and  Whitehead  and  Russell,  has  de- 
pended very  largely  upon  a  further  study  of  the  logic  of  relations.  While 
we  can  do  no  more,  within  reasonable  limits,  than  to  suggest  the  manner 
of  this  development,  it  seems  best  that  the  most  important  of  these  proper- 
ties of  relations  should  be  given  in  outline.  But  even  this  outline  cannot 
be  complete,  because  the  theoretical  basis  provided  by  our  previous  dis- 
cussion is  not  sufficient  for  completeness. 

Every  relation,  .R,  has  a  converse,  «R,  which  can  be  defined  as  follows: 

17-01     vR  =  yx(xRy}.        Def. 

If  x  has  the  relation  R  to  y,  then  y  has  the  converse  relation,  «R,  to  ar. 

It  follows  at  once  from  the  definition  of  (xmy^)  e.  R  that 

xmRyn  =  yn  *R  xm 

because  (xm  R  yn)  =  (xmyn]  e  R  =  (ynxm)  e  "R  =  yn  *R  xm. 
The  converse  of  the  converse  of  R  is  R. 

«(«#)  =  R 
19 


274  A  Survey  of  Symbolic  Logic 

since    «(«#)  =  xy(y^Rx)  =  xy(xRy}  =  R.     (This    is    not    a    proof: 
proof  would  require  that  we  demonstrate 

n,,  „[(*,  y)  *  w(wfl)  =  (x,  y)  e  R] 

But  it  is  obvious  that  such  a  demonstration  may  be  given.  In  general, 
we  shall  not  pause  for  proofs  here,  but  merely  indicate  the  method  of  proof.) 
The  properties  of  symmetrical  relations  follow  from  the  theorems  con- 
cerning converses.  For  any  symmetrical  relation  T,  T  =  ^T.  The  uni- 
versal relation,  1,  and  the  null-relation,  0,  are  both  symmetrical: 


(x  1  y}  =  [f(.r,  y}  cf(z,  y)]  =  1  =  [f(y,  x)  cf(y,  x)}  =  (y  1  x) 

(The  "1"  in  the  middle  of  this  'proof  is  obviously  that  of  the  calculus  of 
propositions.     Similarly  for  0  in  the  next.) 

(x  0  y)  =  -[$(x,  y}  c  $(X)  y}]  =  0  =  -[f  (y,  x)  c  f  (</,  x}]  =  (y  0  x) 

It  is  obvious  that  if  two  relations  are  equivalent,  their  converses  will  be 
equivalent: 

(#   =    £)    =    (w#    =    wfi[) 

Not  quite  so  obvious  is  the  equivalent  of  (RcS),  in  terms  of  «R  and  ^S. 
We  might  expect  that  (RcS)  would  give  (*S  c^>R).     Instead  we  have 

(R  c  S)  =  (v/fl  c  wS) 

for    (fi  c  S)  =  nx,  „[(*  Ry)c(xS  y)]  =  Iix,  y[(y  *R  .T)  c  (y  wfi  «)] 

=  (»Rc»S) 

"'Parent  of  implies  'ancestor  of"  is  equivalent  to  '"Child  of  implies 
'descendent  of". 

The  converses  of  compound  relations  is  as  folio  WTS: 


for   x»(RxS)y  =  y(R*S)x  =  (y  R  x}  x  (y  S  x}  =  (x^Ry)  x(x»Sy) 

=  x(^R  x  * 

If  x  is  employer  and  exploiter  of  y,  the  relation  of  y  to  x  is  "employee  of 
and  exploited  by".     Similarly 

v(R  +  S)  =  vR  +  »S 

If  x  is  either  employer  or  benefactor  of  y,  the  relation  of  y  to  x  is  "either 
employee  of  or  benefitted  by". 

Other  important  properties  of  relations  concern  "relative  sums"  and 


Systems  Based  on  Material  Implication  275 

"relative  products".  These  must  be  distinguished  from  the  non-relative 
sum  and  product  of  relations,  symbolized  by  •*•  and  x  .  The  non-relative 
product  of  "friend  of"  and  "colleague  of"  is  "friend  and  colleague  of": 
their  relative  product  is  "friend  of  a  colleague  of".  Their  non-relative 
sum  is  "either  friend  of  or  colleague  of":  their  relative  sum  is  "friend  of 
every  non-colleague  of  ".  We  shall  denote  the  relative  product  of  R  and  S 
by  R  |  S,  their  relative  sum  by  R  t  S. 

17-02     R\8  =  xz{2y[(xRy)  x(ySz)]}.        Def. 

R\S  is  the  relation  of  the  couple  (x,  z)  when  for  some  y,  x  has  the  relation 
R  to  y  and  y  has  the  relation  S  to  z.  x  is  friend  of  a  colleague  of  z  when, 
for  some  y,  x  is  friend  of  y  and  y  is  colleague  of  z. 

17-03     R-tS  =  xz{Uy[(xRy)  +  (ySz)]}.         Def. 

R  t  S  is  the  relation  of  x  to  z  when,  for  every  y,  either  x  has  the  relation 
R  to  y  or  y  has  the  relation  S  to  z.  x  is  friend  of  all  non-colleagues  of  z 
when,  for  every  y,  either  x  is  friend  of  y  or  y  is  colleague  of  z. 

It  is  noteworthy  that  neither  relative  products  nor  relative  sums  are 
commutative.  "Friend  of  a  colleague  of"  is  not  "colleague  of  a  friend  of". 
Nor  is  "friend  of  all  non-colleagues  of"  the  same  as  "colleague  of  all  non- 
friends  of".  But  both  relations  are  associative. 

R\(S\T)  =  (R\S}\T 

for         ?x{(wRx}  x[x(S  T)z}}  =  2x{(w  Rx)  x?y[(x  S  y}  x(y  T  z)]} 

=  2y2x{(wRx)x((xSy)x(yTz)}} 
=  S,S«{[(w£aOx(a:S30]x(3fr2)} 

=  2y{2x[(wRx)x(xSz)]x(y  T  z}} 
=  2y{(w(RS)y]x(yTz)} 

"Friend  of  a  (colleague  of  a  neighbor  of)"  is  "(friend  of  a  colleague)  of  a 
neighbor  of". 

Similarly,  R  t  (S  t  T)  =  (R  t  S)  t  T 

"Friend  of  all  (non-colleagues  of  all  non-neighbors  of)"  is  "(friend  of  all 
non-colleagues)  of  all  non-neighbors  of". 

De  Morgan's  Theorem  holds  for  the  negation  of  relative  sums  and  prod- 
ucts. 


for  -M(x  R  y)  x  (y  S  z)]  }  =  Uy  -[(x  Ry)x(yS  z)} 


276  A  Survey  of  Symbolic  Logic 

The  negative  of  "friend  of  a  colleague  of"  is  "non-friend  of  all  colleagues 

(non-non-colleagues)  of  ". 

Similarly,  -(R  t  S)  =  -R\-S 

The  negative  of  "friend  of  all  non-colleagues  of"  is  "non-friend  of  a  non- 
colleague  of  ". 

Converses  of  relative  sums  and  products  are  as  follows: 


for  x  *(R  |  S)z  =  z(R  \  S)x  =  Sv[(z  Ry)x(yS  x)] 

=  Zy((ySx)x(zRy)} 


If  x  is  employer  of  a  benefactor  of  z,  then  the  relation  of  z  to  x  is  "bene- 

fitted  by  an  employee  of". 

Similarly,  »(R  t  S)  =  *S  t  ~R 

If  x  is  hater  of  all  non-helpers  of  z,  the  relation  of  z  to  x  is  "  helped  by  all 
who  are  not  hated  by". 

The  relation  of  relative  product  is  distributive  with  reference  to  non- 
relative  addition. 

R\(S+  T}  =  (RS)  +  (R  T) 

for          x[R  \(S+T)}z  =  Zv{(xRy)x  [y(S  +  7>]  } 


Similarly,  (R  +  S)   T  =  (R  \  T)  +  (S  \  T) 

"Either  friend  or  colleague  of  a  teacher  of"  is  the  same  as  "either  friend 
of  a  teacher  of  or  colleague  of  a  teacher  of". 
A  somewhat  curious  formula  is  the  following: 


It  holds  since     x[R  \  (S  x  T)]z  =  2y{(x  Ry)x  [y(S  x  7>]  } 

=  Zy{(xRy)x[(ySz)x(yTz)}\ 

and  since  a  x  (b  xc)  =  (a  x6)  x  (a  xc), 


c  2,[(X  Ry)x(yS  z)}  x  2,[(x  Ry}x(yT  z}] 
And  this  last  expression  is  [x(R\S)z]  x[x(R\  T)z]. 


Systems  Based  on  Material  Implication  277 

If  x  is  student  of  a  friend  and  colleague  of  z,  then  x  is  student  of  a  friend  and 
student  of  a  colleague  of  z.  The  converse  implication  does  not  hold,  be- 
cause "student  of  a  friend  and  colleague"  requires  that  the  friend  and  the 
colleague  be  identical,  while  "student  of  a  friend  and  student  of  a  col- 
league" does  not.  (Note  the  last  step  in  the  'proof ',  where  Sv  is  repeated, 
and  observe  that  this  step  carries  exactly  that  significance.) 

Similarly,  (R  x  S)  \  T  c  (R  \  T)  x  (S  \  T) 

The  corresponding  formulae  with  t  instead  of  |  are  more  complicated 
and  seldom  useful;  they  are  omitted. 

The  relative  sum  is  of  no  particular  importance,  but  the  relative  product 
is  a  very  useful  concept.  In  terms  of  this  idea,  "powers"  of  a  relation  are 
definable : 

R2  =  R    R,        R3  =  R*    R,        etc. 

A  transitive  relation,  S,  is  distinguished  by  the  fact  that  *S2  c  S,  and  hence 
Sn  c  S.  The  predecessors  of  predecessors  of  predecessors  ...  of  x  are 
predecessors  of  x.  This  conception  of  the  powers  of  a  relation  plays  a 
prominent  part  in  the  analysis  of  serial  order,  and  of  the  fundamental  proper- 
ties of  the  number  series.  By  use  ofthis  and  certain  other  concepts,  the 
method  of  "mathematical  induction"  can  be  demonstrated  to  be  com- 
pletely deductive.20 

In  the  work  of  De  Morgan  and  Peirce,  "relative  terms"  were  not  given 
separate  treatment.  The  letters  by  which  relations  were  symbolized  were 
also  interpreted  as  relative  terms  by  a  sort  of  systematic  ambiguity.  Any 
relation  symbol  also  stood  for  the  class  of  entities  which  have  that  relation 
to  something.  But  in  the  logistic  development  of  mathematics,  since  that 
time,  notably  in  Principia  Mathematical  relative  terms  are  given  the 
separate  treatment  which  they  really  require.  The  "domain"  of  a  given 
relation,  R — that  is,  the  class  of  entities  which  have  the  relation  R  to  some- 
thing or  other — may  be  symbolized  by  D'R,  which  can  be  defined  as  follows: 

17-04     D'R  =  x[-2y(xRy)}.        Def. 

The  domain  of  R  is  the  class  of  .r's  determined  by  the  function  "For  some  y, 
x  has  the  relation  R  to  y".  If  R  be  "employer  of",  D'R  will  be  the  class 
of  employers. 

The  "converse  domain"  of  R — that  is,  the  class  of  things  to  each  of 

20  See  Principia,  i,  Bk.  n,  Sect.  E. 

21  See  i,  ^33.     The  notation  we  use  for  domains  and  converse  domains  is  that  of  Prin- 
cipia. 


278  A  Survey  of  Symbolic  Logic 

which  something  or  other  has  the  relation  R — may  be  symbolized  by  Q.'R 
and  similarly  defined: 

17-05     (Tfl  =  $[2x(xRy)}.        Def. 

The  converse  domain  of  R  is  the  class  of  y's  determined  by  the  function 
"For  some  x,  x  has  the  relation  R  to  y".  If  R  be  "employer  of",  Q'R 
will  be  the  class  of  employees. 

The  domain  and  converse  domain  of  a  relation,  R,  together  constitute 
the  "field  "of  R,  C'R. 

17-06    C'R  =  £{2y[(xRy)  +  (yRx)]}. 

The  field  of  R  will  be  the  class  of  all  terms  which  stand  in  either  place  in 
the  relation.  If  R  be  "employer  of",  C'R  is  the  class  of  all  those  who  are 
either  employers  or  employees. 

The  elementary  properties  of  such  "relative  terms"  are  all  obvious: 

xn  e  T>'R  =  ?y(xn  R  y} 
yn*a'R  =  2x(xRyn~) 
xn  e  C'R  =  -2y[(xn  Ry)  +  (yR  xn}} 
C'R  =  D'R  +  d'R 

However,  for  the  logistic  development  of  mathematics,  these  properties 
are  of  the  highest  importance.  We  quote  from  Principia  Mathematical 22 
"  Let  us  ...  suppose  that  R  is  the  sort  of  relation  that  generates  a  series, 
say  the  relation  of  less  to  greater  among  integers.  Then  D'R  =  all  integers 
that  are  less  than  some  other  integer  =  all  integers,  Q'R  =  all  integers 
that  are  greater  than  some  other  integer  =  all  integers  except  0.  In  this 
case,  C'.R  =  all  integers  that  are  either  greater  or  less  than  some  other 
integer  =  all  integers  ....  Thus  when  R  generates  a  series,  C'R  becomes 
important.  ..." 

We  have  now  surveyed  the  most  fundamental  and  important  characters 
of  the  logic  of  relations,  and  we  could  not  well  proceed  further  without 
elaboration  of  a  kind  which  is  here  inadmissible.  But  the  reader  is  warned 
that  we  have  no  more  than  scratched  the  surface  of  this  important  topic. 
About  1890,  Schroder  could  write  "What  a  pity!  To  have  a  highly 
developed  instrument  and  nothing  to  do  with  it".  And  he  proceeded  to 
make  a  beginning  in  the  bettering  of  this  situation  by  applying  the  logic 
of  relatives  to  the  logistic  development  of  certain  portions  of  Dedekind's 
theory  of  number.  Since  that  time,  the  significance  of  symbolic  logic  has 
been  completely  demonstrated  in  the  development  of  Peano's  Formulaire 

22 1,  p.  261. 


Systems  Based  on  Material  Implication  279 

and  of  Principia  Mathematica.  And  the  very  head  and  front  of  this  develop- 
ment is  a  theory  of  relations  far  more  extended  and  complete  than  any 
previously  given.  We  can  here  adapt  the  prophetic  words  which  Leibniz 
puts  into  the  mouth  of  Philalethes :  "  I  begin  to  get  a  very  different  opinion 
of  logic  from  that  which  I  formerly  had.  I  had  regarded  it  as  a  scholar's 
diversion,  but  I  now  see  that,  in  the  way  you  understand  it,  it  is  a  kind  of 

universal  mathematics." 

\ 

VI.     THE  LOGIC  OF  Principia  Mathematica 

We  have  now  presented  the  extensions  of  the  Boole-Schroder  Algebra — 
the  Two-Valued  Algebra,  propositional  functions  and  the  propositions 
derived  from  them,  and  the  application  to  these  of  the  laws  of  the  Two- 
Valued  Algebra,  giving  the  calculus  of  propositional  functions.  Beyond 
this,  we  have  shown  in  outline  how  it  is  possible,  beginning  with  the  Two- 
Valued  Algebra  as  a  calculus  of  propositions,  to  derive  the  logic  of  classes 
in  a  form  somewhat  more  satisfactory  than  the  Boole-Schroder  Algebra, 
and  the  logic  of  relations  and  relative  terms.  In  so  doing,  we  have  presented 
as  much  of  that  development  which  begins  with  Boole  and  passes  through 
the  work  of  Peirce  to  Schroder  as  is  likely  to  be  permanently  significant. 
But,  our  purpose  here  being  expository  rather  than  historical,  we  have  not 
followed  the  exact  forms  which  that  development  took.  Instead,  we  have 
considerably  modified  it  in  the  light  of  what  symbolic  logicians  have  learned 
since  the  publication  of  the  work  of  Peirce  and  Schroder. 

Those  who  are  interested  to  note  in  detail  our  divergence  from  the 
historical  development  will  be  able  to  do  so  by  reference  to  Sections  VII 
and  VIII  of  Chapter  I.  But  it  seems  best  here  to  point  out  briefly  what 
these  alterations  are  that  we  have  made.  In  the  first  place,  we  have 
interpreted  2<px,  U<px,  2$(x,  y),  etc.,  explicitly  as  sums  or  products  of 
propositions  of  the  form  <pxn,  \}/(xmyn},  etc.  Peirce  and  Schroder  avoided 
this,  in  consideration  of  the  serious  theoretical  difficulties.  But  while 
they  did  not  treat  II  <px  as  an  actual  product,  2<px  as  an  actual  sum,  still 
the  laws  which  they  give  for  propositions  of  this  type  are  those  which  result 
from  such  a  treatment.  There  is  no  slightest  doubt  that  the  method  by 
which  Peirce  discovered  and  formulated  these  laws  is  substantially  the  one 
which  we  have  exhibited.  And  this  explicit  use  of  II <px  as  the  symbol  for  a 
product,  ~L<f>x  as  the  symbol  of  a  sum,  makes  demonstration  possible  where 
otherwise  a  large  number  of  assumptions  must  be  made  and,  for  further 
principles,  a  much  more  difficult  and  less  obvious  style  of  proof  resorted  to. 


280  A  Survey  of  Symbolic  Logic 

In  this  part  of  their  work,  Peirce  and  Schroder  can  hardly  be  said  to  have 
formulated  the  assumptions  or  given  the  proofs. 

In  the  second  place,  the  Boole-Schroder  Algebra — the  general  outline 
of  which  is  already  present  in  Peirce 's  work — probably  seemed  to  Peirce 
and  Schroder  an  adequate  calculus  of  classes  (though  there  are  indications 
in  the  paper  of  1880  that  Peirce  felt  its  defects).  With  this  system  before 
them,  they  neglected  the  possibility  of  a  better  procedure,  by  beginning 
with  the  calculus  of  propositions  and  deriving  the  logic  of  classes  from  the 
laws  which  govern  prepositional  functions.  And  although  the  principles 
which  they  formulate  for  prepositional  functions  are  as  applicable  to  func- 
tions of  one  as  of  two  variables,  and  are  given  for  one  as  well  as  for  two, 
their  interest  was  almost  entirely  in  functions  of  two  and  the  calculus  of 
relatives  which  may  be  derived  from  such  functions.  The  logic  of  classes 
which  we  have  outlined  is,  then,  something  which  they  laid  the  foundation 
for,  but  did  not  develop. 

The  main  purposes  of  our  exposition  thus  far  in  the  chapter  have  been 
two:  first,  to  make  clear  the  relation  of  this  earlier  treatment  of  symbolic 
logic  with  the  later  and  better  treatment  to  be  discussed  in  this  section; 
and  second,  to  present  the  logic  of  prepositional  functions  and  their  deriva- 
tives in  a  form  somewhat  simpler  and  more  easily  intelligible  than  it  might 
otherwise  be.  The  theoretically  sounder  and  more  adequate  logic  of  Prin- 
cipia  Mathematica  is  given  a  form  which — so  far  as  prepositional  functions 
and  their  derivatives  is  concerned — seems  to  us  to  obscure,  by  its  notation, 
the  obvious  and  helpful  mathematical  analogies,  and  requires  a  style  of 
proof  which  is  much  less  obvious.  With  regard  to  this  second  purpose,  we 
disclaim  any  idea  that  the  development  we  have  given  is  theoretically 
adequate;  its  chief  value  should  be  that  of  an  introductory  study,  prepara- 
tory to  the  more  complex  and  difficult  treatment  which  obviates  the  the- 
oretical shortcomings. 

Incidentally,  the  exposition  which  has  been  given  will  serve  to  indicate 
how  much  we  are  indebted,  for  the  recent  development  of  our  subject,  to 
the  earlier  work  of  Peirce  and  Schroder. 

The  Peirce-Schroder  symbolic  logic  is  closely  related  to  the  logic  of 
Peano's  Formulaire  de  Mathematiques  and  of  Principia  Mathematica.  This 
connection  is  easily  overlooked  by  the  student,  with  the  result  that  the  sub- 
ject of  his  first  studies — the  Boole-Schroder  Algebra  and  its  applications — is 
likely  to  seem  quite  unrelated  to  the  topic  which  later  interests  him — the 
logistic  development  of  mathematics.  Both  the  connections  of  these  two 


Systems  Based  on  Material  Implication  281 

and  their  differences  are  important.  We  shall  attempt  to  point  out  both. 
And  because,  for  one  reason,  clearness  requires  that  we  stick  to  a  single 
illustration,  our  comparison  will  be  between  the  content  of  preceding 
sections  of  this  chapter  and  the  mathematical  logic  of  Book  I,  Prindpia 
Mathematical 

The  Two- Valued  Algebra  is  a  calculus  produced  by  adding  to  and  re- 
interpreting an  algebra  intended  primarily  to  deal  with  the  relations  of 
classes.  And  it  has  several  defects  which  reflect  this  origin.  In  the  first 
place,  the  same  logical  relation  is  expressed,  in  this  system,  in  two  different 
ways.  We  have,  for  example,  the  proposition  "If  p  c q  and  q  c  r,  then 
per",  where  p,  q,  and  r  are  propositions.  But  "if  .  .  .  ,  then  ..." 
is  supposed  to  be  the  same  relation  which  is  expressed  by  c  in  p  c  q,  q  c  r, 
and  per.  Also,  "and"  in  "peg  and  q  c  r  "  is  the  relation  which  is  other- 
wise expressed  by  x— and  so  on,  for  the  other  logical  relations.  The 
system  involves  the  use  of  "if  .  .  .  ,  then  .  .  .",  "...  and  .  .  .", 
"either  .  .  .  or  .  .  .",  ".  .  .is  equivalent  to  .  .  .",  and  "...  is  not 
equivalent  to  .  .  .",  just  as  any  mathematical  system  may;  yet  these  are 
exactly  the  relations  c ,  x ,  +  ,  = ,  and  =1=  whose  properties  are  supposed  to 
be  investigated  in  the  system.  Thus  the  system  takes  the  laws  of  the  logical 
relations  of  propositions  for  granted  in  order  to  prove  them.  Nor  is  this 
paradox  removed  by  the  fact  that  we  can  demonstrate  the  interchange- 
ability  of  "if  .  .  .  ,  then  ..."  and  c,  of  ".  .  .  and  ..."  and  x,  etc. 
For  the  very  demonstration  of  this  interchangeability  takes  for  granted 
the  logic  of  propositions;  and  furthermore,  in  the  system  as  developed, 
it  is  impossible  in  most  cases  to  give  a  law  the  completely  symbolic  form 
until  it  has  first  been  proved  in  the  form  which  involves  the  non-symbolic 
expression  of  relations.  So  that  there  is  no  way  in  which  the  circularity 
in  the  demonstration  of  the  laws  of  propositions  can  be  removed  in  this 
system. 

Another  defect  of  the  Two-Valued  Algebra  is  the  redundance  of  forms. 
The  proposition  p  or  "p  is  true"  is  symbolized  by  p,  by  p  =  1,  by  p  =(=  0, 

23  Logically,  as  well  as  historically,  the  method  of  Peano's  Formulaire  is  a  sort  of 
intermediary  between  the  Peirce-Schroder  mode  of  procedure  and  Prindpia.  The  general 
method  of  analysis  and  much  of  the  notation  follows  that  of  the  Formulaire.  But  the 
Formulaire  is  somewhat  less  concerned  with  the  extreme  of  logical  rigor,  and  somewhat 
more  concerned  with  the  detail  of  the  various  branches  of  mathematics.  Perhaps  for  this 
reason,  it  lacks  that  detailed  examination  and  analysis  of  fundamentals  which  is  the  dis- 
tinguishing characteristic  of  Prindpia.  For  example,  the  Formulaire  retains  the  ambiguity 
of  the  relation  a  (in  our  notation,  c ):  p  D<?  may  be  either  "the  class  p  is  contained  in 
the  class  q",  or  "the  proposition  p  implies  the  proposition  q".  In  consequence,  the  Formu- 
laire contains  no  specific  theory  of  propositions. 


282  A  Survey  of  Symbolic  Logic 

etc.,  the  negation  of  p  or  "p  is  false"  by  -p,  p  =  0,  -p  =  1,  p  4=  1,  etc. 
These  various  forms  may,  it  is  true,  be  reduced  in  number;  p  and  -p  may 
be  made  to  do  service  for  all  their  various  equivalents.  But  these  equivalents 
cannot  be  banished,  for  in  the  proofs  it  is  necessary  to  make  use  of  the  fact 
that  p  =  (p  =  1)  =  (p  4=  0),  -p  =  (p  =  0)  =  (-p  =  1),  etc.,  in  order  to 
demonstrate  the  theorems.  Hence  this  redundance  is  not  altogether 
avoidable. 

Both  these  defects  are  removed  by  the  procedure  adopted  for  the 
calculus  of  propositions  in  Principia  Mathematical  Here  p  =  1,  p  =  0, 
etc.,  are  not  used;  instead  we  have  simply  p  and  its  negative,  symbolized 
by  ~p.  And,  impossible  as  it  may  seem,  the  logic  of  propositions  which 
every  mathematical  system  has  always  taken  for  granted  is  not  presumed. 
The  primitive  ideas  are:  (1)  elementary  propositions,  (2)  elementary 
prepositional  functions,  (3)  assertion,  (4)  assertion  of  a  prepositional  func- 
tion, (5)  negation,  (6)  disjunction,  or  the  logical  sum;  and  finally,  the 
idea  of  "equivalent  by  definition",  which  does  not  belong  in  the  system 
but  is  merely  a  notation  to  indicate  that  one  symbol  or  complex  of  symbols 
may  be  replaced  by  another.  An  elementary  proposition  is  one  which 
does  not  involve  any  variables,  and  an  elementary  prepositional  function 
is  such  as  "not-p"  where  p  is  an  undetermined  elementary  proposition. 
The  idea  of  assertion  is  just  what  would  be  supposed — a  proposition  may 
be  asserted  or  merely  considered.  The  sign  |-  prefaces  all  propositions 
which  are  asserted.  An  asserted  prepositional  function  is  such  as  "A  is  A  " 
where  A  is  undetermined.  The  disjunction  of  p  and  q  is  symbolized  by 
p  v  q,  instead  of  p  +  q.  p  v  q  means  "  At  least  one  of  the  two  propositions, 
p  and  q,  is  true  ". 

The  postulates  and  definitions  are  as  follows: 

#1-01     psg.  =  .~pvg.         Df. 

"p  (materially)  implies  q"  is  the  defined  equivalent  of  "At  least  one  of 
the  two,  'p  is  false'  and  'q  is  true',  is  a  true  proposition".  (The  explana- 
tion of  propositions  here  is  ours.)  p  3  q  is  the  same  relation  which  we 
have  symbolized  by  peg,  not  its  converse. 

(The  propositions  quoted  wrill  be  given  the  number  which  they  have  in 
Principia.  The  asterisk  which  precedes  the  number  will  distinguish 
them  from  our  propositions  in  earlier  chapters  or  earlier  sections  of  this 
chapter.) 

The  logical  product  of  p  and  q  is  symbolized  by  p  q,  or  p  .  g. 

24  See  Bk.  i,  Sect.  A. 


Systems  Based  on  Material  Implication  283 

*3-01     p.g.  =  .  ~(~pv~g).          Df. 

"p  is  true  and  q  is  true"  is  the  defined  equivalent  of  "It  is  false  that  at 
least  one  of  the  two,  p  and  q,  is  false".  This  is,  of  course,  a  form  of  De 
Morgan's  Theorem — in  our  notation,  (p  q)  =  -(-p  +  -q). 

The  (material)  equivalence  of  p  and  q  is  symbolized  by  p  =  q  or  p  .  =  .  q. 

*4-01     p  =  q,  =  m  poq.q?  p.         Df. 

"p  is  (materially)  equivalent  to  q"  is  the  defined  equivalent  of  "p  (ma- 
terially) implies  q  and  q  (materially)  implies  p".  In  our  notation,  this 
would  be  (p  =  q)  =  (p  cg)(q  cp).  Note  that  ...  =  ...  and  ...  =  ... 
Df  are  different  relations  in  Principia. 

The  dots  in  these  definitions  serve  as  punctuation  in  place  of  parentheses 
and  brackets.  Two  dots,  :,  takes  precedence  over  one,  as  a  bracket  over  a 
parenthesis,  three  over  two,  etc.  In  *4-01  we  have  only  one  dot  after  =, 
because  the  dot  between  p  3  q  and  q  3  p  indicates  a  product :  a  dot,  t>r  two 
dots,  indicating  a  product  is  always  inferior  to  a  stop  indicated  by  the 
same  number  of  dots  but  not  indicating  a  product. 

The  postulates  of  the  system  in  question  are  as  follows: 

#1-1     Anything  implied  by  a  true  elementary  proposition  is  true.         Pp. 
("Pp."  stands  for  "Primitive  proposition".) 

*1  •  11  When  <px  can  be  asserted,  where  x  is  a  real  variable,  and  <px  3  \f/x 
can  be  asserted,  where  a:  is  a  real  variable,  then  i/x  can  be  asserted,  where  x 
is  a  real  variable.  Pp. 

A  "real  variable"  is  such  as  p  in  -p. 
*l-2       f-rpvp.o.p.         Pp. 

In  our  notation,  (p  +  p*)  c  p. 
*l-3       |-:g.3.pvg.         Pp. 

In  our  notation,  q  c  (p  +  q). 
*l-4       |-:  p  vg  .  3  .  q  vp.         Pp. 

In  our  notation,  (p  +  q)  c  (q  +  p). 
*l-5       h!  p  v  (q  vr)  .  3.  q  v  (p  vr).         Pp. 

In  our  notation,  [p  +  (q  +  r}}  c  [q  +  (p  +  r)]. 

*l-6       h!.<7=>r.D:pvg.D.pvr.         Pp. 
In  our  notation,  (qcr)  c  [(p  +  q)  c(p  +  r)]. 
Note  that  the  sign  of  assertion  in  each  of  the  above  is  followed  by  a 


284  A  Survey  of  Symbolic  Logic 

sufficient  number  of  dots  to  indicate  that  the  whole  of  what  follows  is 
asserted. 

#1-7  If  p  is  an  elementary  proposition,  ~p  is  an  elementary  proposition. 
Pp. 

#1-71  If  p  and  q  are  elementary  propositions,  pvq  is  an  elementary 
proposition.  Pp. 

#1-72  If  <pp  and  \f/p  are  elementary  prepositional  functions  which  take 
elementary  propositions  as  arguments,  <pp  v  \f/p  is  an  elementary  preposi- 
tional function.  Pp. 

This  completes  the  list  of  assumptions.  The  last  three  have  to  do 
directly  with  the  method  by  which  the  system  is  developed.  By  *l-7, 
any  proposition  which  is  assumed  or  proved  for  p  may  also  be  asserted  to 
hold  for  ~p,  that  is  to  say,  ~p  may  be  substituted  for  p  or  q  or  r,  etc.,  in 
any  proposition  of  the  system.  By  #1-71,  p  vq  may  be  substituted  for  p 
or  q  or  r,  etc.  And  by  *1  •  72,  if  any  two  complexes  of  the  foregoing  symbols 
which  make  sense  as  "statements"  can  be  treated  in  a  certain  way  in  the 
system,  their  disjunction  can  be  similarly  treated.  By  the  use  of  all  three 
of  these,  any  combination  such  as  pvq,  p  .  q,  p^q,  p  =  q,  p^q*q^P, 
~p .  v .  p  v  q,  ~j)  v  ~q,  etc.,  etc.,  may  be  substituted  for  p  or  q  or  r  in  any 
assumed  proposition  or  any  theorem.  Such  substitution,  for  which  no 
postulates  would  ordinarily  be  stated,  is  one  of  the  fundamental  operations 
by  which  the  system  is  developed. 

Another  kind  of  substitution  which  is  fundamental  is  the  substitution 
for  any  complex  of  symbols  of  its  .defined  equivalent,  where  such  exists. 
This  operation  is  covered  by  the  meaning  assigned  to  "...  =  ...  Df ". 

Only  one  other  operation  is  used  in  the  development  of  this  calculus 
of  elementary  propositions — the  operation  for  which  #1-1  and  #1-11  are 
assumed.  If  by  such  substitutions  as  have  just  been  explained  there 
results  a  complex  of  symbols  in  which  the  main,  or  asserted,  relation  is  3 , 
and  if  that  part  of  the  expression  which  precedes  this  sign  is  identical  with  a 
postulate  or  previous  theorem,  then  that  part  of  the  expression  which 
follows  this  sign  may  be  asserted  as  a  lemma  or  new  theorem.  In  other 
wrords,  a  main,  or  asserted,  sign  D  has,  by  *!•!  and  *1-11,  the  significant 
property  of  "If  .  .  .  ,  then  .  .  .".  This  property  is  explicitly  assumed 
in  the  postulates.  The  main  thing  to  be  noted  about  this  operation  of 
inference  is  that  it  is  not  so  much  a  piece  of  reasoning  as  a  mechanical,  or 
strictly  mathematical,  operation  for  which  a  rule  has  been  given.  No 


Systems  Based  on  Material  Implication  285 

"mental"  operation  is  involved  except  that  required  to  recognize  a  previous 
proposition  followed  by  the  main  implication  sign,  and  to  set  off  what 
follows  that  sign  as  a  new  assertion.  The  use  of  this  operation  does  not, 
then,  mean  that  the  processes  and  principles  of  ordinary  logic  are  tacitly 
presupposed  as  warrant  for  the  operations  which  give  proof. 

What  is  the  significance  of  this  assumption  of  the  obvious  in  *1  •  1,  •#•!  •  11, 

#1-7,  *1-71,  and  #1-72?  Precisely  this:  these  postulates  explicitly 
assume  so  much  of  the  logical  operations  as  is  necessary  to  develop  the 
system,  and  beyond  this  the  logic  of  propositions  simply  is  not  assumed. 
To  illustrate  this  fact,  it  will  be  well  to  consider  carefully  an  exemplary 
proof  or  two. 

#2-01      h:p=>~;p  .3-  ~;p 

r      ~P~I 

Dem.       Taut—     \-i  ~p  v  ~p  .  3  .  ~p        (1) 

[(!).(*!•  01)]  hipD-p.D.-p 

"Taut"  is  the  abbreviation  for  the  Principle  of  Tautology,  *l-2  above. 
~plp  indicates  that  ~p  is  substituted  in  this  postulate  for  p,  giving  (1). 
This  operation  is  valid  by  *l-7.  Then  by  the  definition  #1-01,  above, 
p  D  ~p  is  substituted  for  its  defined  equivalent,  ~p  v  ~p,  and  the  proof  is 
complete. 

*2-05       H  •  q^r  .2:  poq  .D.  p^r 

Dem.        Sum  —  \  \-z  .  qor  .0  '  ~p  vo  .  D  .  ~p  vr         (1) 
L          p  J 


[(1)  .  (-&1-01)]  \-l  .  q^r  .D:  poq  ,D.  por 

Here  "Sum"  refers  to  ^1-6,  above.  And  (1)  is  what  ^1-6  becomes  when 
~p  is  substituted  for  p.  Then,  by  *1-01,  p^>q  and  p^r  are  substituted 
for  their  defined  equivalents,  ~pvg  and  ~pvr,  in  (1),  and  the  resulting 
expression  is  the  theorem  to  be  proved. 

The  next  proof  illustrates  the  use  of  *1  •  1  and  *1  •  11. 

•&2-06       h:  •  p^q  -3!  q^r  .3.  por 


Dem. 

p 


r'  ?3?'  "-'I 
,         q,         r     J 


3r.3:j>Dg'.D.p3r!.3:.;pDqr.D:gor.3.p3r    (1) 

[#2-05]  h!  •  q^r.  ?:p^q.  om  por  (2) 

[(1)  .  (2)  .  #1-11]  \-:  .  poq  .0:  qor  .o.p^r 


286  A  Survey  of  Symbolic  Logic 

" Comm "  is  #2 •  04,  previously  proved,  which  is  p.o.q^riosqo.por. 
When,  in  this  theorem,  q  3  r  is  substituted  for  p,  p  3  q  for  q,  and  p  3  r  for  r, 
it  becomes  the  long  expression  (1).  Such  substitutions  are  valid  by  *l-7, 
#1-71,  and  the  definition  *1-01:  if  p  is  a  proposition,  ~p  is  a  proposition; 
if  ~p  and  q  are  propositions,  ~p  v  q  is  a  proposition ;  and  p  3  q  is  the  defined 
equivalent  of  ~p  v  q.  Thus  poq  can  be  substituted  for  p.  If  we  replace 
the  dots  by  parentheses,  etc.,  (1)  becomes 

h  { (g  3  r)  3  [(p  3  g)  3  (p  3  r)] }  3  { (p  3  g)  3  [(q  3  r)  3  (p  3  r)} } 

But,  as  (2)  states,  what  here  precedes  the  main  implication  sign  is  identical 
with  a  previous  theorem,  ^2-05.  Hence,  by  *1-11,  what  follows  this 
main  implication  sign — the  theorem  to  be  proved — can  be  asserted. 

Further  proofs  would,  naturally,  be  more  complicated,  but  they  involve 
no  principle  not  exemplified  in  the  above.  These  three  operations — sub- 
stitutions according  to  *l-7,  *1-71,  and  *l-72;  substitution  of  defined 
equivalents;  and  "inference"  according  to  #1-1  and  *1-11 — are  the  only 
processes  which  ever  enter  into  any  demonstration  in  the  logic  of  Principia. 
The  result  is  that  this  development  avoids  the  paradox  of  taking  the  logic 
of  propositions  for  granted  in  order  to  prove  it.  Nothing  of  the  sort  is 
assumed  except  these  explicitly  stated  postulates  whose  use  we  have  ob- 
served. And  it  results  from  this  mode  of  development  that  the  system  is 
completely  symbolic,  except  for  a  few  postulates,  *1  •  1,  *1  •  7,  etc.,  involving 
no  further  use  of  "if  .  .  .  ,  then  .  .  .",  "either  ...  or  ...","...  and 
"  etc 

•         •         •  j      wlfVft 

We  have  now  seen  that  the  calculus  of  propositions  in  Principia  Mathe- 
matica  avoids  both  the  defects  of  the  Two-Valued  Algebra.  The  further 
comparison  of  the  two  systems  can  be  made  in  a  sentence :  Except  for  the 
absence,  in  the  logic  of  Principia,  of  the  redundance  of  forms,  p,  p  =  1, 
p  =t=  0,  etc.,  etc.,  and  the  absence  of  the  entities  0  and  1,  the  two  systems 
are  identical.  Any  theorem  of  this  part  of  Principia  can  be  translated 
into  a  valid  theorem  of  the  Two-Valued  Algebra,  and  any  theorem  of  the 
Two-Valued  Algebra  not  involving  0  and  1  otherwise  than  as  {  =  0 }  or 
{  =  1 }  can  be  translated  into  a  valid  theorem  of  Principia.  In  fact,  the 
qualification  is  not  particularly  significant,  because  any  use  of  0  and  1  in 
the  Two- Valued  Algebra  reduces  to  their  use  as  {  =  0 }  and  {  =  1 } .  For  0 
as  a  term  of  a  sum,  and  1  as  a  factor,  immediately  disappear,  while  the 
presence  of  0  as  a  factor  and  the  presence  of  1  in  a  sum  can  always  be  other- 
wise expressed.  But  p  =  0  is  -p,  and  p  —  I  is  p.  Hence  the  two  systems 


Systems  Based  on  Material  Implication  287 

are  simply  identical  so  far  as  the  logical  significance  of  the  propositions 
they  contain  is  concerned.25 

The  comparison  of  our  treatment  of  propositional  functions  with  the 
same  topic  in  Principia  is  not  quite  so  simple.26 

In  the  first  place,  there  is,  in  Principia,  the  "theory  of  types,"  which 
concerns  the  range  of  significance  of  functions.  But  we  shall  omit  con- 
sideration of  this.  Then,  there  are  the  differences  of  notation.  Where 
we  write  TL<px,  or  Hx(px,  Principia  has  (x)  .  <px;  and  where  we  write  2<px, 
or  2x<px,  Principia  has  (3#)  .  <px.  A  further  and  more  important  difference 
may  be  made  clear  by  citing  the  assumptions  of  Principia. 

*9-01  ~{(x)  .  <px}  .  =  .  (3a;)  .  ~<px.  Df. 
*9-02  ~{(3z)  .  <px}  .  =  .  0)  .  ~<px.  Df. 
*9-03  (X)  .  <px  .  vp  :  =  .  (x)  .  <px  vp.  Df. 

25  This  may  be  proved  by  noting  that,  properly  translated,  the  postulates  of  each  system 
are  contained  amongst  the  propositions  of   the  other.     Of   the    postulates  in  Principia, 
rendered  in  our  notation: 

•^1-01  is  (pc.q)  =  (-p  +  q),  which  is  contained  in  our  theorem  9-3. 

^1-2  is  (p  +  p)  cp,  which  is  a  consequence  of  our  theorems  2-2  and  5-33. 

^•1-3  is,  p  c  (p  +  q},  which  is  our  theorem  5-21. 

^•1-4  is  (p  +  q)  c  (q  +  p),  which  follows  from  our  theorem  4-3,  by  2-2. 

^1-5  is,  p  +  (q  +  r)  cq  +  (p  +  r),  which  is  a  consequence  of  our  theorems  4-3  and  4-4, 

by  2-2. 

*l-6  is  (q  cr)  c  [(p  +  q)  c  (p  +  r)],  which  is  a  consequence  of  our  theorem  5-31,  by  2-2. 
The  remaining  (non-symbolic)  postulates  are  tacitly  assumed  in  our  system. 
Of  our  postulates,  1-1-1-9  in  Chap,  n  and  9-01  in  Chap,  iv: 

1-1  is  a  consequence  of  ^-1-7  and  ^-1-71  in  Principia. 

1-2  is  ->K4'24  in  Principia. 

1-3  is  ^-4-3  in  Principia. 

1  -4  is  ^2-3  in  Principia. 

1  -5  is  equivalent  to  "If  a;  =  0,  then  a  x  =  0",  hence  to  -x  c  -(a  x),  which  is  a  consequence 

of  *3-27  in  Principia,  by  *2-16. 

1-61,  in  the  form  -(x  -a)  c  (x  a  =  x),  is  a  consequence  of  ^-4 -71  and  ^4-61  in  Principia, 

by  *4-01  and  *3-26. 

1-62,  in  the  form   [(y  a  =  y}(y  -a  =  y)]c-y,  is  a  consequence  of  ^-4-71,   ^-5-16,  and 

•^2-21  in  Principia. 

1-7  is  equivalent  to  [(x  =  \)(y  =  0)]  c  (x  =  -y),  hence  to  (x  -y}  c  (x  =  -y),  which  is  an 

immediate  consequence  of  ^5-1  in  Principia. 

1  •  8  is  ^4  •  57  in  Principia. 

1-9  is  ^4-71  in  Principia. 

9-01  is  equivalent  to  (q  =  1)  c  [p  =  (p  =  q)],  hence  to  q  c  [p  =  (p  =  q)],  which  is  an 

immediate  consequence  of  ^5-501  in  Principia. 

26  See  Principia,  i,  15-21. 


288  A  Survey  of  Symbolic  Logic 

In  this  last,  note  the  difference  in  the  scope  of  the  "quantifier"  (x)  on  the 
two  sides.     If  the  dots  be  replaced  by  parentheses,  *9  •  03  will  be 

{[(*)  -  (px]  vp}  =  {(x)  ,[<pxvp]\ 

A  similar  difference  in  the  scope  of  (x)  or  (3.r)  on  the  two  sides  characterizes 
each  of  the  further  definitions.27 

*9  •  04  p  .  v .  (x)  .  <px  :  =  .  (x)  .  p  v  <px.        Df . 

•#•9  •  05  (3.r)  .  <px  .  v .  p  :  =  .  (3x)  .  <px  v  p.        Df . 

*9  -06  p  .  v .  (3.r)  .  (px  :  =  .  (3.r)  .  p  v  <px.        Df. 

*9-07  (x)  .  (px  .  v .  (%)  .  ^  :  =  :  (x)  :  (3y)  .  #e  vty.        Df. 

*9  -08  (3y)  .  fy  .  v .  (ar)  .  <px  :  =  :  (x)  :  (3#)  .  ^  v  <px.        Df. 

Besides  these  definitions,  there  are  four  postulates  (in  addition  to  those 
which  underlie  the  calculus  of  elementary  propositions). 

*9-l      |-:  <px  •  3 .  (3  2) .  tpz.        Pp. 

*9  •  1 1      H!  <px  v  <py  .  D  .  (3  2)  .  <pz.        Pp. 

^9-12     What  is  implied  by  a  true  premiss  is  true.         Pp. 

*9-13     In  any  assertion  containing  a  real  variable,  this  real  variable  may 

be  turned  into  an  apparent  variable  for  which  all  possible  values  are  asserted 

to  satisfy  the  function  in  question.         Pp. 

By  our  method,  every  one  of  these  assumptions,  except  *9-12,  is  a 
proved  proposition.  In  our  notation, 

*9-01  is  -II  (px  =  S  -<px,  which  is  our  theorem  10-1,  with  -<px  substituted 
for  <px. 

#9-02  is  -2<px  =  Tl-<px,  which  is  our  theorem   10-12,  with  -<px  substi- 
tuted for  (px. 

*9-03  is  Ti(px  +  P  =  Ux((px  +  P),  which  is  our  theorem  10-32. 

*9-04  is  P  +  n<Ar  =  nz(P+  (px),  which  is  10-33. 

*9-05  is  2(px  +  P  =  Zx(>.r  +  P),  which  is  10-3. 

*9-06  is  P+  Z(px  =  SZ(P+  (px),  which  is  10-31. 

*9-07  is  H<px  +  2\l/y  =  H.x'2y((px  +  -*l/y),  which  is  contained  in  12-5. 

*9-08  is  2^y  +  Hpx  =  nx2y(if/y  •»•  (px),  which  is  also  contained  in  12-5. 

The  postulates  require  explanation.  The  authors  of  Principia  use 
<py,  <pz,  etc.,  to  represent  values  of  the  function  (px.  In  other  words,  where 
we  have  written  (pxn  they  simply  change  the  letter.  This  is  a  valid  con- 

27  Ibid.,  i,  135-38. 


Systems  Based  on  Material  Implication  289 

vention  (though  it  often  renders  proofs  confusing)  because  the  range  of  <px 
is  determined  by  <p,  not  by  x,  and  x  is — conventions  aside — indifferent. 
z  in  <pz,  where  we  should  write  <pxn,  is  called  a  "real  variable",  x  in  (x)  .  tpx 
and  (3.r)  .  px,  an  "apparent  variable".  With  this  explanation,  it  is  clear 
that: 

#9-1  is   <pxnc.*L<px,  which  is  10-21. 

#9-11  is  (pxm+  <pxnc*2<px,  which  is  an  immediate  consequence  of  10-21, 
by  5 -33. 

#9-13  is  "If  whatever  value  of  x,  in  <px,  xn  may  be,  <pxn,  then  H<px,"  and 
this  implication  is  contained  in  the  equivalence  stated  by  10-23. 

These  principles  which  are  assumed  in  Principia  Mathematica  are  suf- 
ficient to  give  all  further  propositions  concerning  functions  of  one  variable, 
without  assuming  (x)  .  <px  to  be  the  product  of  <px\,  <f>x2,  etc.  (or  <py,  <pz, 
etc.),  (3#)  .  <px  to  be  the  sum  of  <pxi,  <f>Xz,  etc.  These  are  simply  assumed 
as  new  primitive  ideas,  (x)  .  <px  meaning  "  <px  for  all  values  of  x",  (3ar)  .  <px 
meaning  "  <px  for  some  values  of  x".  This  procedure  obviates  all  questions 
about  the  number  of  values  of  x  in  px — which  troubled  us — and  secures 
the  universality  of  theorems  involving  prepositional  functions  without  any 
discussion  or  convention  covering  the  cases  in  which  the  values  of  the  vari- 
able are  infinite  in  number.  The  proofs  in  Principia  reflect  this  difference 
of  method.  They  are,  in  general,  what  ours  might  have  been  if  we  had 
based  all  further  proofs  directly  upon  10-23  and  the  propositions  con- 
necting 2<px  +  P  with  21(<f>x  +  P),  etc.,  not  making  any  use,  after  10-23, 
of  the  properties  of  n  <px  as  a  product,  or  of  2<px  as  a  sum. 

The  theory  of  functions  of  two  variables,  in  Principia  Mathematica, 
requires  two  further  assumptions: 

#11-01     (x,  y)  .  <p(x,  ?/).  =  :  GC)  :  (y*)  .  <p(x,  y).        Df. 
*11  -03     (Bar,  y)  .  v(x,  y)  .  =  :  (3x)  :  (%)  .  <p(x,  y).        Df. 
These  are  identically  our  assumptions : 
11-06     nx>  y<p(x,  y)  =  nxUv<p(x,  y),  and 
11-05     SZf  y<p(x,  y}  =  2x2y<p(x,  y). 

The  difference  between  the  treatment  of  prepositional  functions  which 
we  have  given  and  the  treatment  in  Principia  is  not  necessarily  correlated 
with  the  difference  between  our  treatment  of  propositions  and  theirs.  The 
method  by  which  we  have  developed  the  theory  of  propositional  functions 

20 


290  A  Survey  of  Symbolic  Logic 

might  exactly  as  well  have  been  based  upon  the  calculus  of  elementary 
propositions  in  Principia  as  upon  the  Two- Valued  Algebra.  A  few  minor 
alterations  would  be  sufficient  for  this  change.  The  different  procedure 
for  prepositional  functions,  in  the  two  cases,  is  a  difference  to  be  adjudged 
independently,  without  necessary  reference  to  the  defects  of  the  Two- 
Valued  Algebra  which  have  been  pointed  out. 

Beyond  the  important  differences  which  have  been  mentioned,  there 
are  minor  and  trivial  divergences  between  the  two  systems,  due  to  the 
different  use  of  notation.  Neglecting  these,  we  may  say  that  the  two 
methods  give  the  same  results,  with  the  following  exceptions: 

1.  There  are  certain  complexities  in  Principia  due  to  the  theory  of 
types. 

2.  In  Principia  the  conditions  of  significance  are  explicitly  investigated. 

3.  Principia  contains  a  theory  of  "descriptions",  account  of  which  is 
here  omitted. 

But  none  of  these  exceptions  is  a  necessary  difference.  They  are  due  to 
the  more  elementary  character  of  our  presentation  of  the  subject.  We 
may,  then,  say  loosely  that  the  two  methods  give  identical  results. 

The  calculus  of  classes  and  of  relations  which  we  have  outlined  in  the 
preceding  sections  bear  a  similar  relation  to  the  logic  of  classes  and  of 
relations  in  Principia;  that  is  to  say,  there  is  much  more  detail  and  com- 
plexity of  theory  in  Principia,  but  so  far  as  our  exposition  goes,  the  two  are 
roughly  the  same.  And  here  there  is  no  important  difference  of  method. 

It  should  now  be  clear  how  the  logic  of  Principia  is  related  to  the  logic 
we  have  presented,  following  in  the  main  the  methods  of  Peirce  and  Schroder. 
There  is  much  difference  of  method,  and,  especially  in  the  case  of  the  cal- 
culus of  propositions,  this  difference  is  in  favor  of  Principia.  And  in 
Principia  there  is  much  more  of  theoretical  rigor  and  consequent  complexity : 
also  there  are  important  extensions,  especially  in  the  theory  of  "descrip- 
tions" and  the  logic  of  relatives.  But  so  far  as  the  logic  which  we  have 
expounded  goes,  the  two  methods  give  roughly  identical  results.  When 
we  remember  the  date  of  the  work  of  Peirce  and  Schroder,  it  becomes  clear 
what  is  our  debt  to  them  for  the  better  developments  which  have  since 
been  made. 


CHAPTER    V 
THE  SYSTEM  OF  STRICT  IMPLICATION1 

The  systems  discussed  in  the  last  chapter  were  all  based  upon  material 
implication,  peg  meaning  exactly  "The  statement,  ' p  is  true  and  q  false/ 
is  a  false  statement".  We  have  already  called  attention  to  the  fact  that 
this  is  not  the  usual  meaning  of  "implies".  Its  divergence  from  the 
"implies"  of  ordinary  inference  is  exhibited  in  such  theorems  as  "A  false 
proposition  implies  any  proposition",  and  "A  true  proposition  is  implied 
by  any  proposition".2 

The  present  chapter  intends  to  present,  in  outline,  a  calculus  of  propo- 
sitions which  is  based  upon  an  entirely  different  meaning  of  "  implies  "- 
one  more  in  accord  with  the  customary  uses  of  that  relation  in  inference 
and  proof.  We  shall  call  it  the  system  of  Strict  Implication.  And  we  shall 
refer  to  Material  Implication,  meaning  either  the  Two-Valued  Algebra  or 
the  calculus  of  propositions  as  it  appears  in  Principia  Mathematica,  since 
the  logical  import  of  these  two  systems  is  identical.  It  will  appear  that 
Strict  Implication  is  neither  a  calculus  of  extensions,  like  Material  Impli- 
cation and  the  Boole-Schroder  Algebra,  nor  a  calculus  of  intensions,  like 
the  unsuccessful  systems  of  Lambert  and  Castillon.  It  includes  relations 
of  both  types,  but  distinguishes  them  and  shows  their  connections.  Strict 
Implication  contains  Material  Implication,  as  it  appears  in  Principia 
Mathematica,  as  a  partial-system,  and  it  contains  also  a  supplementary 
partial-system  the  relations  of  which  are  those  of  intension. 

The  numerous  questions  concerning  the  exact  significance  of  implication, 
and  the  ordinary  or  "proper"  meaning  of  "implies",  will  be  discussed  in 
Section  V. 

It  will  be  indicated  how  Strict  Implication,  by  an  extension  to  preposi- 
tional functions,  gives  a  calculus  of  classes  and  class-concepts  which  exhibits 
their  relations  both  in  extension  and  in  intension.  In  this,  it  provides  the 

1  Various  studies  toward  this  system  have  appeared  in  Mind  and  the  Journal  of  Phi- 
losophy (see  Bibliography).     But  the  complete  system  has  not  previously  been  printed. 
We  here  correct,  also,  certain  errors  of  these  earlier  papers,  most  notably  with  reference  to- 
triadic  "strict"  relations. 

2  For  further  illustrations,  see  Chap,  n,  Sect,  i,  and  Lewis,  "Interesting  Theorems  ia 
Symbolic  Logic,"  Jour.  Philos.,  Psych.,  etc.  x  (1913),  p.  239. 

291 


292  A  Survey  of  Symbolic  Logic 

calculus  of  intensions,  so  often  attempted  before,  so  far  as  such  a  calculus 
is  possible  at  all. 

I.    PRIMITIVE  IDEAS,  PRIMITIVE  PROPOSITIONS,  AND  IMMEDIATE  CONSE- 
QUENCES 

• 

The  fundamental  ideas  of  the  system  are  similar  to  those  of  MacColl's 
Symbolic  Logic  and  its  Applications.  They  are  as  follows: 

1.  Propositions:  p,  q,  r,  etc. 

2.  Negation:  -p,  meaning  "p  is  false". 

3.  Impossibility:  ~p,  meaning  "p  is  impossible",  or  "It  is  impossible 
that  p  be  true".3 

4.  The  logical  product:  p  x</  or  p  q,  meaning  "p  and  q  both",  or  "p  is 
true  and  q  is  true". 

5.  Equivalence:  p  =  q,  the  defining  relation. 

Systems  previously  developed,  except  MacColl's,  have  only  two  truth- 
values,  "true"  and  "false".  The  addition  of  the  idea  of  impossibility 
gives  us  five  truth- values,  all  of  which  are  familiar  logical  ideas: 

(1)  p,  "pis  true". 

(2)  -p,  "pis  false". 

(3)  ~p,  "p  is  impossible". 

(4)  — p,  "It  is  false  that  p  is  impossible" — i.  e.,  "p  is  possible". 

(5)  ~-p,  "It  is  impossible  that  p  be  false" — i.  e.,  "p  is  necessarily 
true". 

Strictly,  the  last  two  should  be  written  -(~p)  and  ~(-p):  the  parentheses 
are  regularly  omitted  for  typographical  reasons. 

The  reader  need  be  at  no  pains  to  grasp  — p  and  ~-p  as  simple  ideas: 
it  is  sufficient  to  understand  -p  and  ~p,  and  to  remember  that  each  such 
prefix  affects  the  letter  as  already  modified  by  those  nearer  it.  It  should 
be  noted  that  there  are  also  more  complex  truth-values.  — p  is  equivalent 

to  p,  as  will  be  shown,  but p,  ~  -  ~p,  -  ~  — p,  etc.,  are  irreducible. 

We  shall  have  occasion  to  make  use  of  only  one  of  these, p,  "  It  is 

false  that  it  is  impossible  that  p  be  true" — i.  e.,  "p  is  possibly  false".4 
Each  one  of  these  complex  truth-values  is  a  distinct  and  recognizable  idea, 
though  they  are  seldom  needed  in  logic  or  in  mathematics. 

*  We  here  use  a  symbol,  ~,  which  appears  in  Principia  Mathematical  with  a  different 
meaning.  The  excuse  for  this  is  its  typographical  convenience. 

4  MacColl  uses  a  single  symbol  for  — p,  "p  is  possibly  true"  and  -~-p,  "p  is  possibly 
false". 


The  System  of  Strict  Implication  293 

The  dyadic  relations  of  propositions  can  be  defined  in  terms  of  these 
truth-values  and  the  logical  product,  p  q. 

1-01     Consistency.     poq  =  -~(pq).         Def. 

~(pq)>  "ft  is  impossible  that  p  and  q  both  be  true"  would  be  "p  and  q 
are  inconsistent".  Hence  -~(pq),  "It  is  possible  that  p  and  q  both  be 
true",  represents  " p  and  q  are  consistent". 

1-02  Strict  Implication.     p-*q  =  ~(p-q)-         Def. 

1-03  Material  Implication.     pcq  =  -(p-q).         Def. 

1-04  Strict  Logical  Sum.     p*q  =  ~(-p~~q)'         Def. 

1-05  Material  Logical  Sum.     p+q  =  ~(~p~q)-         Def. 

1-06  Strict  Equivalence,     (p  =  q)  =  (p  -iq)(q-*  P)  •         Def. 

We  here  define  the  defining  relation  itself,  because  by  this  procedure  we. 
establish  the  connection  between  strict  equivalence  and  strict  implication. 
Also,  this  definition  makes  it  possible  to  deduce  expressions  of  the  type, 
p  =  q — something  which  could  not  otherwise  be  done.5  But  p  =  q  re- 
mains a  primitive  idea  as  the  idea  that  one  set  of  symbols  may  be  replaced 
by  another. 

1  •  07     Material  Equivalence,     (p  =  q)  =  (p  c  q)  (qcp*).         Def. 

These  eight  relations — the  seven  defined  above  and  the  primitive  rela- 
tion, p  q — divide  into  two  sets,  p  q,  p  c  q,  p  +  q,  and  p  =  q  are  the  relations 
which  figure  in  any  calculus  of  Material  Implication.  We  shall  refer  to 
them  as  the  "material  relations",  p  o  q,  p-lq,  p*q,  and  p  =  q  involve 
the  idea  of  impossibility,  and  do  not  belong  to  systems  of  Material  Impli- 
cation. These  may  be  called  the  "strict  relations".  We  may  anticipate  a 
little  and  exhibit  the  analogy  of  these  two  sets,  which  results  from  the 
theorem 

~(p  q}  =  -(P  °  1} 

shortly  to  be  proved. 

Strict  relations :  Material  relations : 

p  -J  q  =  -(p  O  -q)  p  c  q  =  -(p  -q) 

p  A  q  =  -(-p  o  -<?)  P  +  Q  =  -(-p  -q) 

(p  =  q)  =  -(p  o  -q}  *  -(q  °  -p)  (p  =  q)  =  -(p  -q)  *-(?  -p) 

B  The  "circularity"  here  belongs  inevitably  to  logic.  No  mathematician  hesitates  to 
prove  the  equivalence  of  two  propositions  by  showing  that  "  If  theorem  A,  then  theorem  B, 
and  if  theorem  B,  then  theorem  A  ".  But  to  do  this  he  must  already  know  that  a  reciprocal 
"if  ...  then  .  .  . "  relation  is  equivalent  to  an  equivalence.  And  the  italicized  "  equivalent 
to"  represents  a  relation  which  must  be  assumed. 


294  A  Survey  of  Symbolic  Logic 

The  reader  will,  very  likely,  have  some  difficulty  in  distinguishing  in  meaning 
p  -J  q  from  p  c  q,  p  A  q  from  p  +  q.  The  above  comparison  may  be  of  assist- 
ance in  this  connection,  since  it  translates  these  relations  in  terms  of  p  o  q 
and  p  q.  We  shall  be  in  no  danger  of  confusing  p  o  q,  "pis  consistent  with 
q,"  with  p  q,  "p  and  q  are  both  true". 

Both  p  A  q  and  p  +  q  would  be  read  "  Either  p  or  q  ".  But  p  A  q  denotes 
a  necessary  connection;  p+  q  a  merely  factual  one.  Let  p  represent  "To-*, 
day  is  Monday",  and  q,  "2  +  2=4".  Then  p+q  is  true  but  p*q  is 
false.  In  point  of  fact,  at  least  one  of  the  two  propositions,  "Today  is 
Monday"  and  "2  +  2  =  4",  is  true;  but  there  is  no  necessary  connection 
between  them.  "Either  .  .  .  or  .  .  ."  is  ambiguous  in  this  respect.  Ask 
the  members  of  any  company  whether  the  proposition  "Either  today  is 
Monday  or  2  +  2  =  4"  is  true,  and  they  will  disagree.  Some  will  confine 
"Either  ...  or  ..."  to  the  p*q  meaning,  others  will  make  it  include 
the  p+q  meaning;  few,  or  none,  will  make  the  necessary  distinction. 

Similarly,  the  difference  between  p  =  q  and  p  =  q  is  that  p  =  q  denotes 
an  equivalence  of  logical  import  or  meaning,  while  p  =  q  denotes  simply 
an  equivalence  of  truth-value.  As  was  shown  in  Chapter  II,  p  =  q  may  be 
accurately  rendered  "p  and  q  are  both  true  or  both  false".  Here  again, 
the  strict  relation,  p  =  q,  symbolizes  a  necessary  connection;  the  material 
relation,  p  =  q,  a  merely  factual  one. 

The  postulates  of  the  system  are  as  follows : 

1-1     p  q-4  q  p 

If  p  and  q  are  both  true,  then  q  and  p  are  both  true. 

1'2     qp-*p 

If  q  and  p  are  both  true,  then  p  is  true. 

1-3    p  -lp p 

If  p  is  true,  then  p  is  true  and  p  is  true. 
1-4     p(qr)4q(pr) 

If  p  is  true  and  q  and  r  are  both  true,  then  q  is  true  and  p  and  r  are  both 
true. 

1-S    p-i-(-p) 

If  p  is  true,  then  it  is  false  that  p  is  false. 

1-6    (pHgXgHr)-j(p-ir) 

If  p  strictly  implies  q  and  q  strictly  implies  r,  then  p  strictly  implies  r. 


The  System  of  Strict  Implication  295 

1-7    ~p*-p 

If  it  is  impossible  that  p  be  true,  then  p  is  false. 

1-8     p  -1  g  =  ~g  H  ~p 

"p  strictly  implies  q"  is  equivalent  to  "'q  is  impossible'  strictly  im- 
plies 'p  is  impossible'". 

The  first  six  of  these  present  no  novelty  except  the  relation  -i .  They 
do  not,  so  far,  distinguish  this  system  from  Material  Implication.  But, 
as  we  shall  see  shortly,  the  postulates  1  •  7  and  1  •  8  are  principles  of  trans- 
formation; they  operate  upon  the  other  postulates,  and  on  themselves, 
and  thus  introduce  the  distinguishing  characteristics  of  the  system.  Postu- 
late 1  -7  is  obvious  enough.  Postulate  1  -8  is  equivalent  to  the  pair, 

(p  -J  q)  -J  (-  ~p  -J  -  ~g)     If  p  implies  q,  then  '  p  is  possible '  implies 

'q  is  possible'. 
(~p  -J  ~g)  -J  (-p  -J  -q)       If  'p  is  impossible'  implies  'q  is  impossible', 

then  'p  is  false'  implies  ' q  is  false'. 

These  two  propositions  are  more  "self-evident"  than  the  postulate,  but 
they  express  exactly  the  same  relations. 

(To  eliminate  parentheses,  as  far  as  possible,  we  make  the  convention 
that  the  sign  =,  unless  in  parentheses,  takes  precedence  over  any  other 
relation ;  that  -i  and  c  take  precedence  over  A  ,  + ,  o ,  and  x  ;  that  A 
and  +  take  precedence  over  o  and  x ;  and  that  •*  takes  precedence  over 
c .  Thus 

pq  +  -p-q*pcq         is          [(p  q)  +  (-p  -q)]  -1  (p  eg) 
and          p  c  q  r  =  (p  c  g)  (p  c  r)         is         [p  c  (q  r)]  =  [(p  c  q)  (p  c  r)] 

However,  where  there  is  a  possibility  of  confusion,  we  shall  put  in  the 
parentheses.) 

The  operations  by  which  theorems  are  to  be  derived  from  the  postulates 
are  three: 

1.  Substitution. — Any  proposition  may  be  substituted  for  p  or  g  or  r, 
etc.     If  p  is  a  proposition,  -p  and  ~p  are  propositions.     If  p  and  g  are 
propositions,  p  q  is  a  proposition.     Also,  of  any  pair  of  expressions  related 
by  = ,  either  may  be  substituted  for  the  other. 

2.  Inference. — If  p  is  asserted  and  p  n  g  is  asserted,  then  g  may  be 
asserted.     (Note  that  this  operation  is  not  assumed  for  material  impli- 
cation, p  c  g.) 

3.  Production. — If  p  and  g  are  separately  asserted,  p  q  may  be  asserted. 
These  are  the  only  operations  made  use  of  in  proof. 


296  A  Survey  of  Symbolic  Logic 

In  order  to  make  clearer  the  nature  of  the  strict  relations,  and  particu- 
larly strict  implication,  we  shall  wish  to  derive  from  the  postulates  their 
correlates  in  terms  of  strict  relations.  This  can  be  done  by  the  use  of 
postulate  1-8  and  its  consequences,  for  by  1-8  a  relation  of  two  material 
relations  can  be  transformed  into  a  relation  of  the  corresponding  strict 
relations.  But  as  a  preliminary  to  exhibiting  this  analogy,  we  must  prove  a 
number  of  simple  but  fundamental  theorems.  These  working  principles 
will  constitute  the  remainder  of  this  section. 

The  first  theorem  will  be  proved  in  full  and  the  proof  explained.  The 
conventions  exemplified  in  this  proof  are  used  throughout. 

2-  1     p  q  -J  p 

1-6  {pq/p;  qp/q;  p/r}:     1-1  xl-2  -J  (p  q  -ip) 

This  proof  may  be  read  :  "  Proposition  1  •  6,  when  p  q  is  substituted  for  p, 
qp  for  q,  and  p  for  r,  states  that  propositions  1-1  and  1-2  together  imply 
(p  q  -t  p)  ".  The  number  of  the  proposition  which  states  any  line  of  proof 
is  given  at  the  beginning  of  the  line.  Next,  in  braces,  is  indication  of  any 
substitutions  to  be  made.  "  p  q/p"  indicates  that  p  q  is  to  be  substituted, 
in  the  proposition  cited,  for  p;  "p  +  q/r"  would  indicate  that  p+  q  was  to 
be  substituted  for  r,  etc.  Suppose  we  take  proposition  1  •  6,  which  is 


and  make  the  substitutions  indicated  by   {p  q/p;  qp/p;  p/r}.     We  then 
get 


This  is  the  expression  which  follows  the  brace  in  the  above  proof.  But 
since  p  q  H  q  p  is  1  •  1,  and  q  p  -J  p  is  1-2,  we  write  1-1  xl-2  instead  of 
(p  q-i  qp)(qp  -ip).  This  calls  attention  to  the  fact  that  what  precedes 
the  main  implication  sign  is  the  product  of  two  previous  propositions. 
Since  1-1  and  1-2  are  separately  asserted,  their  product  may  be  asserted; 
and  since  this  product  may  be  asserted,  what  it  implies  —  the  theorem  to  be 
proved  —  may  be  asserted.  The  advantage  of  this  way  of  writing  the  proofs 
is  its  extreme  brevity.  Yet  anyone  who  wishes  to  reconstruct  the  demon- 
stration finds  here  everything  essential. 

2-11    (p  =  q)  •*  (p  *  q) 

2-1    {p-lq/p;  q-lp/q}:   (p  H  q)  (q  -*  p)  -J  (p  H  q) 
1-06:  (p  =  q)  =  (p-lq)(q-lp) 


The  System  of  Strict  Implication  297 

2-12     (p  =  q)-i(q-ip) 

Similar  proof,  1-2  instead  of  2-1. 

2-2     (p-tq)4(~q-*~p} 

1-06:  1-8  =  [(p*q)-l(~q-l~p}}((~q-l~p}-l(p*q)}  (I) 

2-1:  (1H  Q.E.D. 

In  this  last  proof,  we  introduce  further  abbreviations  of  proof  as  follows: 
(1),  or  (2),  etc.,  is  placed  after  a  lemma  which  has  been  established,  and 
thereafter  in  the  same  proof  we  write  (1),  or  (2),  etc.,  instead  of  that  lemma. 
Also,  we  shall  frequently  write  "Q.E.D."  in  the  last  line  of  proof  instead 
of  repeating  the  theorem  to  be  proved.  In  the  first  line  of  this  proof,  the 
substitutions  which  it  is  necessary  to  make  in  order  to  get 

1  -8  =  [(p  -J  q)  -I  (~?  -i  ~p)][(~g  -J  ~p)  4  (p  -J  g)] 

are  not  indicated  because  they  are  obvious.  And  in  the  second  line,  state- 
ment of  the  required  substitutions  is  omitted  for  the  same  reason.  Such 
abbreviations  will  be  used  frequently  in  later  proofs. 

Theorem  2-2  is  one  of  the  implications  contained  in  postulate  1-8. 
By  the  definition,  1  •  6,  any  strict  equivalence  may  be  replaced  by  a  pair  of 
strict  implications.  By  postulate  1-2  and  theorem  2-1,  either  of  these 
implications  may  be  taken  separately. 

2-21     (~q  *  ~p)  4  (p  •*  q) 

1-2:  [(1)  in  proof  of  2 -2H  Q.E.D. 

This  is  the  other  implication  contained  in  postulate  1-8. 

2-3     (-P4q~)4(-q4p) 

1-1    {-q/p;  -p/q}:  -q-p-l-p-q  (1) 

2-2   {-q  -pip;  -p  -q/q] :  (1)  H  [~(-p  -q)  4  ~(-q  -p}}  (2) 

1-02:  (2)  =  Q.E.D. 
2-4    p4p 

1-2   {p/q}:  pp-lp  (1) 

1-6:  l-3x(l)^Q.E.D. 
2-5    -(-p)^p 

2-4  {-pip}:  -p-l-p  (1) 

2-3   {-p/q}:  (1)H  Q.E.D. 
2-51     -(-p)  =  p 

1-06:  2-5xl-5  =  Q.E.D. 


298  A  Survey  of  Symbolic  Logic 

2-6     (-p  H  -g)  -J  (q  -J  p) 

2-3  {-g/9| :  (-p-J-gHK-gH?]  (l) 

2-51:  (1)  =  Q.E.D. 

2-61     (p  •* -g)  •*  (g -J -p) 

2-6  {-p/p}:  K-p)-J-gH(g-l-p)  (1) 

2-51:  (1)  =  Q.E.D. 

2-62     (p  H  g)  •*  (-g  H  -p) 

2-61   {-g/g}:  [p  -l  -(-g)]  H  (-g  -J  -p)  (1) 

2-51:  (1)  =  Q.E.D. 

2-63     (png)  =  (-g-i-p) 

1-06:  2-62x2-6  =  Q.E.D. 

2-64    (p-J-g)  =  (g-J-p) 

1-06:  2-61x2-3  =  Q.E.D. 

Theorems  2-3,  2-6,  2-61,  and  2-62  are  the  four  forms  of  the  familiar 
principle  that  an  implication  is  converted  by  changing  the  sign  of  both 
terms. 

2-7     (~p -J  ~g)  •*  (-p  H -g) 

2-21   {p/g;  g/p} :  (~p  H  ~g)  -J  (g  -J  p)  (1) 

2-62   {p/g;  g/p} :  (g  H  p)  •<  (-p  -J  -g)  (2) 

1-6:  (1)  x  (2)  H  Q.E.D. 

2-71     (-p  -{  -g)  H  (~p  •*  ~g) 

2-6:  (-p^-g)H(g Hp)  (1) 

2-2   {g/p;  p/g} :  (g  -J  p)  H  (~p  -J  ~g)  (2) 

1-6:  (1)  x  (2)  H  Q.E.D. 

2-712     (-p-J-g)  =  (~p-J~g) 

1-06:  2-71  x2-7  =  Q.E.D. 
2-72     (~ -p -J  ~ -g)H  (p -i  g) 

2-7   {-p/p;  -g/g} :  (~  -p  -l  ~  -g)  H  [-(-p)  -J  -(-g)]  (1) 

2-51:  (1)  =  Q.E.D. 
2-7J     (p-lg)-{(~-p-{~-g) 

2-71   {-p/p;  -g/g} :  [-(-p)  H  -(-g)]  -l  (~  -p  -J  «  -g)  (1) 

2-51:  (1)  =  Q.E.D. 
2-731     (p-lg)  =  (~-p-J~-g) 

1-06:  2-73x2-72  =  Q.E.D. 
2-74     (P4q)4(-~p4-~q) 

2  -  62  { ~g/p;  ~p/g} :  (~g  -t  ~p)  -J  (-  ~p  H  -  ~g)  (1) 

1-6:  2-2  x(l)-i  Q.E.D. 


The  System  of  Strict  Implication  299 

2-75     (-~p4-~q)4(p4q) 

2-6  [~plp;  ~q/q} :  (-  ~p  -{  -  ~g)  •*  (~g  -J  ~p)  (1) 

1-6:  (1)  x2-2lH  Q.E.D. 
2-76     (png)  =  (--p  .*--,?) 

1-06:  2-74x2-75  =  Q.E.D. 

2-77     (p  •*  q)  =  (-  ~p  -I  -  ~?)  =  (~  -p  -J  ~  -q)  =  (-q  -J  -2?)  =  (~?  ^  ~P) 
2-76  X2-731  x2-63  x2-712  =  Q.E.D. 

"p  implies  q"  is  equivalent  to  '"p  is  possible'  implies  'q  is  possible'"  is 
equivalent  to  "'p  is  necessary'  implies  'q  is  necessary'"  is  equivalent  to 
"fq  is  false'  implies  'p  is  false'"  is  equivalent  to  "'q  is  impossible'  implies 
'p  is  impossible'". 

2-6-2-77  are  various  principles  for  transforming  a  strict  implication. 
These  are  all  summed  up  in  2-77.  The  importance  of  this  theorem  will  be 
illustrated  shortly. 

2-8     pq  =  qp 

1-1    {qfp;  p/q}:  qp-lpq  (1) 

1-06:  l-lx(l)  =  Q.E.D. 
2  •  8 1     p  =  pp 

1-2  {p/q}:  pp-tp  (1) 

1-06:  l-3x(l)  =  Q.E.D. 

2-9     p(qr)  =  q(p  r) 

1-4   {q/p-,  p/q}:  q(pr)-lp(qr)  (1) 

1-06:  l-4x(l)  =  Q.E.D. 

2-91     p(qr)  =  (p  q)r 

2-8:  p(qr)  =  p(r  q) 
2-9:  p(rq)  =  r(p  q) 
2-8:  r(pq)  =  (p  q)r 

The  above  theorems  constitute  a  preliminary  set,  sufficient  to  give 
briefly  most  further  proofs. 

II.    STRICT  RELATIONS  AND  MATERIAL  RELATIONS 

We  proceed  now  to  exhibit  a  certain  analogy  between  strict  relations 
and  material  relations;  between  truths  and  falsities  on  the  one  hand  and 
necessities,  possibilities,  and  impossibilities  on  the  other.  This  analogy 
runs  all  through  the  system:  it  is  exemplified  by  2-77. 


300  A  Survey  of  Symbolic  Logic 

1-1  pq-*qp  3-11  poq-iqop 

If  p  and  q  are  both  true,  then  q  If  p  and  q  are  consistent,  then  q 

and  p  are  both  true.  and  p  are  consistent. 

1-2  qp-lp  3-12  qopl-~p 

If  9  and  p  are  both  true,  then  p  If  g  and  p  are  consistent,  then 

is  true.  it  is  possible  that  p  be  true. 

1-3  p*p  p  3-13  -~p  4  pop 

If  p  is  true,  then  p  and  p  are  If  it  is  possible  that  p  be  true, 

both  true.  then  p  is  consistent  with  itself. 

1-4     p(qr)lq(pr)  3-14     p  o  (q  r)  H  q  o  (p  r) 

The  correspondence  exhibited  in  the  last  line  seems  incomplete.     But 
we  should  note  with  care  that  while 

P(q  r)  =  q(p  r)  =  (p  q)r 

and  any  one  of  these  may  be  read  "p,  q,  and  r  are  all  true",  p  o  (q  or) 
is  not  "p,  q,  and  r  are  all  consistent",  po(qor)  means  "p  is  consistent 
with  the  proposition  'q  is  consistent  with  r'".  Let  p  =  "Today  is  Tues- 
day"; q  =  "Today  is  Thursday";  r  =  "Tomorrow  is  Friday".  Then 
q  o  r  is  true.  And  it  happens  to  be  Tuesday,  so  p  is  true.  Since  p  and 
q  or  are  both  true  in  this  case,  they  must  be  consistent:  p  o  (q  or)  is  true. 
But  "p,  q,  and  r  are  all  consistent"  is  false.  "Today  is  Tuesday"  is  incon- 
sistent with  "Today  is  Thursday"  and  with  "Tomorrow  is  Friday". 
Suppose  we  represent  "p,  q,  and  r  are  all  consistent"  by  p  oq  or.  Then 
as  a  fact,  p  oq  or  will  not  be  equivalent  to  p  o  (q  or).  Instead,  we  shall 
have 

poqor  =  po(qr)  =  q  o  (p  r)  =  (p  q)  or 

"p,  q,  and  r  are  all  consistent"  is  equivalent  to  "p  is  consistent  with  the 
proposition  (q  and  r  are  both  true'",  etc.  We  may,  then,  add  two  new 
definitions : 

3-01    'pqr  =  p(qr).         Def. 
3-02     p  oqor  =  po(qr).         Def. 

3-02  is  typical  of  triadic,  or  polyadic,  strict  relations:  when  parentheses 
are  introduced  into  them,  the  relation  inside  the  parentheses  degenerates 
into  the  corresponding  material  relation.  In  terms  of  the  new  notation  of 
3-01  and  3-02,  the  last  line  of  the  above  table  would  be 

pqrlqpr        poqor-iqopor 
which  exhibits  the  analogy  more  clearly. 


The  System  of  Strict  Implication  301 

We  must  now  prove  the  theorems  in  the  right-hand  column  of  the  table 

3-11     p  oq-i  q  op 

2-74   {qp/p;  qp/q}:  M  •<  [-  ~(p  q)  H  -  ~(g  p)]  (1) 

1-01:  (1)  =  Q.E.D. 
3  •  12  q  o  p  H  -  ~p 

2-74  (gp/p;  p/g}:  1-2  -i  [-  ~(gp)  -J-  ~p]  (1) 

1-01:  (1)  =  Q.E.D. 
3 •  J 3  -  ~p  -1  p  o  p 

2-74  {pp/p}:  1-3H[-~PH-~(PP)]  (1) 

1-01:  (1)  =  Q.E.D. 

3  •  14     p  o  (q  r)  -4  q  o  (p  r) 

2-74   {p(gr)/p-,  q(pr)/q}:  1-4  •«  -  ~[p(q  r)]  -{  -  ~[q(p  r}]  (1) 

1-01:  (1)  =  Q.E.D. 

(In  the  above  proof,  the  whole  of  what  1-4  is  stated  to  imply  should 
be  enclosed  in  a  brace.  But  in  such  cases,  since  no  confusion  will  be  oc- 
casioned thereby,  we  shall  hereafter  omit  the  brace.) 

3  •  15     p  o  (q  r)  =  (p  q)  o  r  =  q  o  (p  r) 

2-76:  2-9H—[p(gr)]  =  -  ~[g(p  r)]  (1) 

2-76:  2-9H-~[p(gr)]  =  -  ~[(p  q)r]  (2) 

1-01:  (2)  x(l)  =  Q.E.D. 

An  exactly  similar  analogy  holds  between  the  material  logical  sum, 
p  +  q,  and  the  strict  logical  sum,  p  A  q. 

3-21  p  +  q-iq  +  p  3-31  p*q4q*p 

"At  least  one  of  the  two,  p  and  "Necessarily  either  p  or  q"  im- 

q,  is  true"  implies  "At  least  one  plies  "Necessarily  either  q  or  p". 

of  the  two,  q  and  p,  is  true". 

3-22  plp  +  q  3-32  ~-p-JpAg 

If  p  is  true,  then  at  least  one  of  If  p  is  necessarily  true,  then 

the  two,  p  and  q,  is  true.  necessarily  either  p  or  q  is  true. 

3-23p  +  p-*p  3-33pApH~-p 

If  at  least  one  of  the  two,  p  and  If  necessarily  either  p  is  true  or 

p,  is  true,  then  p  is  true.  p  is  true,  then  p  is  necessarily  true. 

3-24     p  +  (q  +  r)  H  q  +  (p  +  r)  3-34     p  A  (g  +  r)  -*  </  A  (p  +  r) 

As  before,  the  analogy  in  the  last  line  seems  incomplete,  and  as  before, 
it  really  is  complete.  And  the  explanation  is  similar,  p  +  (q  +  r)  and 


302  A  Survey  of  Symbolic  Logic 

q  +  (p  +  r)  both  mean  "At  least  one  of  the  three,  p,  q,  and  r,  is  true".  But 
p  A  (q  A  r)  would  not  mean  "  One  of  the  three,  p,  q,  and  r,  is  of  necessity 
true".  Instead,  it  would  mean  "One  of  the  two  propositions,  p  and 
'necessarily  either  q  or  r',  is  necessarily  true".  To  distinguish  pA(g  +  r) 
from  p  A  (q  A  r)  is  rather  difficult,  and  an  illustration  just  now,  before  we 
have  discussed  the  case  of  implication,  would  probably  confuse  the  reader. 
We  shall  be  content  to  appeal  to  his  'intuition'  to  confirm  the  fact  that 
"Necessarily  one  of  the  three,  p,  q,  and  r,  is  true"  is  equivalent  to  "Neces- 
sarily either  p  is  true  or  one  of  the  two,  q  and  r,  is  true  " — and  this  last  is 
p  A  (q  +  r).  If  we  chose  to  make  definitions  here,  similar  to  3-01  and  3-02, 
they  would  be 

p  +  q  +  r  =  p  +  (g  +  r) 

and        pA?Ar  =  PA(q  +  r) 
Proof  of  the  theorems  in  the  above  table  is  as  follows: 

3-21     p  +  q-iq  +  p 

1-1    {-q/p;  -p/q}:  -q-p-t-p-q  (1) 

2-62:  (l)H-(-p-g)^-(-?-p)  (2) 

1-05:  (2)  =  Q.E.D. 

3  •  22    p-ip  +  q 

Similar  proof,  using  1  -2  in  place  of  1-1. 

3  •  2  3    p  +  plp 

Similar  proof,  using  1-3. 

3  •  24    p  +  (q  +  r)  -J  q  +  (p  +  r) 

1-4  {-q/p;  -p/q;  -r/r}:  -q(-p  -r)  -J  -p(-q  -r)  (1) 

2  •  62 :  (1)  -J  -[-p(-q  -r)]  -J  -(-q(-p  -r)]  (2) 

2-51:  (2)  =  -{-p-[-(-g-r)]}  H-{-9-[-(-p-r)]}  (3) 

1  •  05 :  (3)  =  p  +  -(-q  -r)  -{  q  +  -(-p  -r)  (4) 

1-05:  (4)  =  Q.E.D. 

3  -  25     p  +  (q  +  r)  =  (p  +  q)  +  r  =  q  +  (p  +  r) 

Similar  proof,  using  2-9  and  2-91,  and  1-06. 

3-31     p  A  g  S  q*p 

1-1    {-q/p;  -p/q}:  -q-p-i-p-q  (1) 

2-2:  (l)H.(-p-g)H-(^r-p)  (2) 

1-04:  (2)  =  Q.E.D. 

3 '  32     ~  -p  -t  p  A  q 

Similar  proof,  using  1  -2  in  place  of  1-1. 


The  System  of  Strict  Implication 


303 


3  -  3 3    p  A  p  -j  ~  -p 

Similar  proof,  using  1-3. 

3  •  34     p  A  (q  +  r)  H  q  A  (p  +  r) 

1-4   {-g/p;  -pfq;  -r/r} :  -q(-p  -r)  -J  -p(-q  -r)  (1) 

2  •  2 :  (1)  H  ~[-p(-q  -r)]  -J  ~[-g(-p  -r)]  (2) 

2-51:  (2)  =  ~{-p-[-(-g-r)]}  .<  ~{-g-[-(-p -r)]}  (3) 

1  •  04 :  (3)  =  p  A  -(-g  -r)  -{  g  A  -(-p  -r)  (4) 

1-05:  (4)  =  Q.E.D. 

3  •  35    p  A  (q  +  r)  =  (p  +  g)  A  r  =  g  A  (p  +  r) 

Similar  proof,  using  2-9  and  2-91,  and  1-06. 

Again,  an  exactly  similar  analogy  holds  between  material  implication, 
peg,  and  strict  implication,  p  -J  q. 

3-41     (pcgH(-gc-p) 

If  p  materially  implies  g,  then  '  q 
is  false'   materially   implies   'p  is 
false'. 
3-42     -p-J(pcg) 

If  p  is  false,  then  p  materially 


2-62      (p-ig^-g-J-p) 

If  p  strictly  implies  g,  then  '  g  is 
false'  strictly  implies  'p  is  false'. 


implies  any  proposition,  g. 

3-43     (pc-p)-j-p 

If  p  materially  implies  its  own 
negation,  then  p  is  false. 

3-44     [pc(gcr)H[gc(pcr)] 


3-52     -p-j(pHg) 

If  p  is  impossible  (not  self-con- 
sistent, absurd),  then  p  strictly  im- 
plies any  proposition,  g. 
3-53     (p-J-p)-l~p 

If  p  strictly  implies  its  own  nega- 
tion, then  p  is  impossible  (not  self- 
consistent,  absurd). 
3-54     [p-J(gcr)H[gn(pcr)] 

The  comparison  of  the  last  line  presents  peculiarities  similar  to  those 
noted  in  previous  tables.  The  significance  of  3-54  is  a  matter  which  can 
be  better  discussed  when  we  have  derived  other  equivalents  of  p  -J  (g  c  r) . 
The  matter  will  be  taken  up  in  detail  further  on. 

The  theorems  of  this  last  table,  like  those  in  previous  tables,  are  got 
by  transforming  the  postulates  1-1,  1-2,  1-3,  and  1-4.  In  consideration 
of  the  importance  of  this  comparison  of  the  two  kinds  of  implication,  we 
may  add  certain  further  theorems  which  are  consequences  of  the  above. 

3-45    p-j(gcp)  3-55     ~-p-l(g-{p) 

If  p  is  true,  then  every  proposi-  If  p  is  necessarily  true,  then  p  is 

tion,  g,  materially  implies  p.  strictly   implied    by   any    proposi- 

tion, g. 


304  A  Survey  of  Symbolic  Logic 

3-46    (-pep)  ip  3-56    (-pnp)-*~-p 

If  p  is  materially  implied  by  its  If  p  is  strictly  implied  by  its  own 

own  denial,  then  p  is  true.  denial,  then  p  is  necessarily  true. 

3-47     -(pcq)-ip  3-57     -(p4q)-l-~p 

If  p  does  not  materially  imply  If  p  does  not  strictly  imply  any 

any  proposition,  q,  then  p  is  true.  proposition,   q,  then  p  is  possible 

(self-consistent) . 

3-48     -(pcg)H-g  3-58     -(p  H  9)  H 9 

If  p  does  not  materially  imply  #,  If  p  does  not  strictly  imply  q, 

then  g  is  false.  then  p  is  possibly  false  (not  neces- 

sarily true). 

Note  that  the  main  or  asserted  implication,  which  we  have  translated 
"If  .  .  .  ,  then  .  .  .",  is  always  a  strict  implication,  in  both  columns. 

3-42  and  3-45-3-48  are  among  the  best  known  of  the  "peculiar"  the- 
orems in  the  system  of  Material  Implication.  For  this  reason,  their  ana- 
logues in  which  the  implication  is  strict  deserve  special  attention.  Let  us 
first  note  that  ~  -p  4  (-p  -J  p)  is  a  special  case  of  3  •  55.  This  and  3  •  56  give 
us  at  once 

-  -P  =  (-P  -*  P) 

This  defines  the  idea  of  "necessity".  A  necessarily  true  proposition — 
e.  g.,  "I  am",  as  conceived  by  Descartes — is. one  whose  denial  strictly 
implies  it.  Similarly,  p-*(-pcp)  is  a  special  case  of  3-45.  And  this, 
with  3-46,  gives 

p  =  (-p  c  p) 

A  true  proposition  is  one  which  is  materially  implied  by  its  own  denial. 
This  point  of  comparison  throws  some  light  upon  the  two  relations. 

The  negative  of  a  necessary  proposition  is  impossible  or  absurd. 
~p  •*  (p  -J  -p)  is  a  special  case  of  3-52.  This,  with  3-53,  gives 

~p  =  (p  H  -p) 

And  p  -{ -p  is  equivalent  to  -(p  o  p).  Thus  an  impossible  or  absurd  propo- 
sition is  one  which  strictly  implies  its  own  denial  and  is  not  consistent 
with  itself.  Correspondingly,  we  get  from  3  •  42  and  3  •  43 

-p  =  (p  c  -p) 

A  false  proposition  is  one  which  materially  implies  its  own  negation. 

It  is  obvious  that  material  implication,  as  exhibited  in  these  theorems, 


The  System  of  Strict  Implication  305 

is  not  the  relation  usually  intended  by  "implies",  but  it  may  be  debated 
whether  the  corresponding  properties  of  strict  implication  are  altogether 
acceptable.     We  shall  revert  to  this  question  later.     At  least,  these  propo- 
sitions serve  to  define  more  sharply  the  nature  of  the  two  relations. 
Proof  of  the  above  theorems  is  as  follows: 

3-41     (pcq)-i(-qc-p) 

1-1    {p/q;  -q/p}:  -q  p  -J  p  -q  (1) 

2-62:  (l)H-(p.g)H-(.?p)  (2) 

2-51:  (2)  =-(p-g)H-[-(/.(-p)]  (3) 

1-03:  (3)  =  Q.E.D. 

3  •  42     -p*(pcq) 

1-2    {p/q;  -q/p}:  p-q-lp  (1) 

2-62:  (IH-p-t-^-g)  (2) 

1-03:  (2)  =  Q.E.D. 

3  •  43     (pc  -p)  -j  -p 

Similar  proof,  using  1  •  3. 

3  •  44     p  c  (q  c  r)  -J  q  c  (p  c  r) 

1-4   {q/p;  p/q;  -r/r}:  q(p  -r)  H  p(q  -r}  (1) 

2-62:  (l)-l-[p(g-r)H-[g(p-r)]  (2) 

2-51:  (2)  =  -{p-[-(g-r)]}  4-{q-[-(p -r)}}  (3) 

1-03:  (3)  =  pc-(q-r)*qc-(p-r)  (4) 

1-03:  (4)  =  Q.E.D. 

3  -  45    p-*(qc-p) 

3-42   \-p/p;  -q/q] :  -(-p}  H  (-pc-q)  (1) 

3.41:  (-pc-g)-j[-(-g)c-(-p)]  (2) 

2-51:  (2)  =  (-pc-g)H(gcp)  (3) 

1-6:  (l)x(3)-i-(-p-)-i(qcp}  (4) 

2-51:  (4)  =  Q.ED. 
3  •  46     (-p  cp~)  lp 

3-43   {-pip}:  [-pc-(-p)H-(-p)  (1) 

2-51:  (1)  =  Q.E.D. 
3-47    -(p  cq)  *p 

2-62   {-pip;  pcq/q}:  3-42  -{  -(p  eg)  ^  -(-p}  (1) 

2-51:  (1)  =  Q.E.D. 
3  -  48     -(p  c  q)  -{  -q 

3-45   {q/p;  pfq}:  q*(pcq)  (1) 

2-62:  (!)H  Q.E.D. 
21 


306  A  Survey  of  Symbolic  Logic 

3-52     ~p4(p*q) 

2-1    {-q/q}:  p-q-tp  (1) 

1-8:  (1)  =  -pH~(p-g)  (2) 

1-02:  (2)  =  Q.E.D. 

3-53     (p-i-p)-l~p 

Similar  proof,  using  1-3. 

3-54     [p-*(qcr)]4[q*(pcr)] 

1-4  {q/p;  p/q;  -r/r}:  q(p -r)  H  p(q -r)  (1) 

1-8:  (1)  =  ~[pfo-r)H-fo(p-r)]  (2) 

2-51:  (2)  =  -{p-Kg-r)]}  -i~{g-[-(p-r)]}  (3) 

1  -02:  (3)  =  [p  H  -(<?  -r)]  -J  [5  •<  -(p  -r)]  (4) 

1-03:  (4)  =  Q.E.D. 

3-55    ~-p-l(g-Jp) 

3  •  52  {-p/p;  -?/?} :  ~  -p  H  (-p  H  -0  (1) 

2-6:  (-pH-g)H(gnp)  (2) 

1-6:  (1)  x  (2)  H  Q.E.D. 

3  •  56     (-p  4  p)  -J  ~  -p 

3-53  {-p/p}:  [-pH-(-p)]H«-p  (1) 

2-51:  (1)  =  Q.E.D. 
3-57  -(p-ig)-«-~p 

2-62:  3-52 -J  Q.E.D. 
3-58  -(p-{g)-J g 

2-62:  3-55 -l  Q.E.D. 

The  presence  of  this  extended  analogy  between  material  relations  and 
strict  relations  in  the  system  enables  us  to  present  the  total  character  of 
the  system  with  reference  to  the  principles  of  transformation,  1-7  and  1-8, 
in  brief  and  systematic  form.  This  will  be  the  topic  of  the  next  section. 

III.      THE   TRANSFORMATION    {-/-I 

We  have  not,  so  far,  considered  any  consequences  of  postulate  1-7, 
"P  -J  -p,  "If  p  is  impossible,  then  p  is  false".  They  are  rather  obvious. 

4  •  1     ~  -p  -J  p 

1-7   {-pip}:  — pH-(-p)  (1) 

2-51:  (1)  =  Q.E.D. 
If  p  is  necessary,  then  p  is  true. 
4  •  12    p-i-~p 

2-61   {-pip;  p/q}:  1-7  -l  Q.E.D. 
If  p  is  true,  then  p  is  possible. 


The  System  of  Strict  Implication  307 

4-13     ~-p-}-~p 

1-6:  4- 1x4- 12 -{Q.E.D. 
If  p  is  necessary,  then  p  is  possible. 
4-14  p  q-lpoq 

4-12   {pq/p}:  pq-*-~(pq)  (1) 

1-01:  (1)  =  Q.E.D. 
4-15     (p-*q)*(pcq) 

1-7  {p-q/p}:  ~(p-g)-f-(p-g)  (1) 

1-02:  (1)  .  (p-*q)-*-(p-q)  (2) 

1-03:  (2)  =  Q.E.D. 
4 '16     p  Ag-j  p  +  q 

1-7  {-p  -g/p} :  -(-P  -q)  4  -(-p  -g)  (1) 

1-04:  (1)  =  pA? *-(-?-?)  (2) 

1-05:  (2)  =  Q.E.D. 
4-17     -(p  o  q)  -{  -(p  g) 

2-62:  4-14  =  Q.E.D. 

By  virtue  of  theorem  4-15,  any  strict  implication  which  is  asserted — 
i.  e.,  is  the  main  relation  in  the  proposition — may  be  replaced  by  a  material 
implication.  And  by  4-16,  any  strict  logical  sum,  A,  which  should  be 
asserted,  may  be  reduced  to  the  corresponding  material  relation,  +  .  The 
case  of  the  strict  relation  "consistent  with",  o,  is  a  little  different.  It 
follows  from  4-17  that  for  every  theorem  in  the  main  relation  o  is  denied, 
that  is,  -(. . .  o  .  .  .),  there  is  an  exactly  similar  theorem  in  which  the  main 
relation  is  that  of  the  logical  product,  that  is,  -(. . .  x  . . .). 

It  is  our  immediate  object  to  show  that  for  every  strict  relation  which  is 
assertable  in  the  system,  the  corresponding  material  relation  is  also  assert- 
able.  It  is,  then,  important  to  know  how  these  various  relations  are  present 
in  the  system.  The  only  relations  so  far  asserted  in  any  proposition  are 
-J  and  = .  Since  =  is  expressible  in  terms  of  -J ,  we  may  take  4  as  the 
fundamental  relation  and  compare  the  others  with  it. 

4-21     p  H  q  =  -p  A  g 

1-02:  p-iq  =  ~(p-q-)  (1) 

2-51:  (1)  =p-,?  =  ~[-(-p)-g]  (2) 

1-04:  (2)  =  Q.E.D. 
4'  22     p  A  g  =  -£M  g 

4-21   {-pip}:  -p-iq  =  -(-p)  Ag  (1) 

2-51:  (1)  =  Q.E.D. 


308  A  Survey  of  Symbolic  Logic 

For  every  postulate  and  theorem  in  which  the  asserted  relation  is  -I , 
there  is  a  corresponding  theorem  in  which  the  asserted  relation  is  A  ,  and 
vice  versa. 

Consider  the  analogous  relations,  c  and  +  . 

4  •  2  3     p  c  q  =  -p  +  q 

1-03:  pcq  =  -(p-q)  (1) 

2-51:  (1)  =  pcq  =  -H-p)-q]  (2) 

1-01:  (2)  =  Q.E.D. 
4-24     p+  q  =  -p  cq 

4-23   {-pip} :  -pcq  =  -(-p)  +  q  (1) 

2-51:  (1)  =  Q.E.D. 

For  every  theorem  in  which  the  asserted  relation  is  c ,  there  is  a  corre- 
sponding theorem  in  wThich  the  asserted  relation  is  + ,  and  vice  versa. 

The  exact  parallelism  between  4-21  and  4-23,  4-22  and  4-24,  corrob- 
orates what  4-16  tells  us:  that  wherever  the  relation  A  is  asserted,  the 
corresponding  material  relation,  +  ,  may  be  asserted. 

4-25     p4q  =  -(po-q) 

1-02:  P4q  =  ~(p-q)  (1) 

2-51:  (1)  =  p*q  =  -[-~(p-q)]  (2) 

1-01:  (2)  =  Q.E.D. 

4-26     poq  =  -(p-*-q) 

4-25   {-q/q}:  p  -J  -q  =  -[p  o  -(-q)]  (1) 

2-51:  (1)  =  p-i-q  =  -(poq-)  (2) 

2-11:  (2H(pH-g)-J-(po?)  (3) 

2-12:  (2)^-(poq)^(p^-q')  (4) 

2-3:  (3)HpogH-(PH-g)  (5) 

2-61:  (4)  -i  -(p  H  -q)  H  p  o  q  (6) 
1-06:  (5)  x(6)  =  Q.E.D. 

For  every  postulate  and  theorem  in  which  the  relation  -i  is  asserted, 
there  is  a  corresponding  theorem  in  which  the  main  relation  is  o  but  this 
relation  is  denied:  and  for  every  possible  theorem  in  which  the  relation  o 
is  asserted,  there  wrill  be  a  corresponding  theorem  in  which  the  main  relation 
is  H  but  that  relation  is  denied,  o  and  •*  are  connected  by  negation. 

An  exact!}7  similar  relation  holds  between  p  q  and  p  cq. 

1-03     pcq  =  -(p-q) 
4-27     pq  =  -(pc-q) 

Proof  similar  to  that  of  4-26,  using  1-03  in  place  of  1-02. 


The  System  of  Strict  Implication  309 

The  parallelism  here  corroborates  4-17:  for  every  possible  theorem  in 
which  the  relation  o  is  denied,  there  is  a  theorem  in  which  the  corre- 
sponding logical  product  is  denied.  But  the  implications  of  4-1^4-17  are 
not  reversible,  and  a  theorem  in  which  the  relation  o  is  asserted  does  not 
give  a  theorem  in  which  any  material  relation  is  asserted.  To  put  it  another 
wray :  of  — p,  p,  and  -  ~p,  the  weakest  is  — p  and  it  cannot  be  further 
reduced.  But  the  truth- value  of  any  consistency  is  [-  ~] — p  oq  =  — (p  q). 

The  reduction  of  =  to  the  corresponding  material  relation,  =,  is  obvious. 

4  •  28     Hypothesis :  p  =  q.     To  prove :  p  =  q. 

2-11:  Hyp.  -J  (p  -i  9)  (1) 

2-12:  Hyp.  H  (q  -J  p)  (2) 

4-15:  (l)-l(pcg)  (3) 

4-15:  (2H(gcp)  (4) 
1-07:  (3)x(4)  =  (p  =  q) 

For  every  theorem  in  which  the  relation  =  is  asserted,  there  is  a  cor 
responding  theorem  in  which  the  relation  =  is  asserted. 

We  have  now  shown  at  length  that,  confining  attention  to  the  main 
relations  in  theorems,  there  are  two  sets  of  strict  relations  which  appear 
in  the  system:  (1)  relations  =,  -{ ,  and  A  which  are  asserted,  and  relations 
o  which  are  denied;  (2)  relations  o  which  are  asserted,  and  relations 
=  ,  -J ,  and  A  which  are  denied.  Wherever  a  relation  of  the  first  described 
set  appears,  it  may  be  replaced  by  the  corresponding  material  relation. 
Any  relation  of  the  second  set  will  be  equivalent  to  some  relation  o  which 
is  asserted — its  truth- value  will  be  [-  •»].  Such  relations  cannot  be  further 
reduced;  they  do  not  give  a  corresponding  material  relation.  But  under 
what  circumstances  will  relations  of  this  second  sort  appear?  Examination 
of  the  postulates  will  show  that  they  can  occur  as  the  main  relation  in  the- 
orems only  through  some  use  of  1  •  7  and  its  consequences,  for  example, 
p  q-ip  o  q,  p  4  — p,  and  ~  -p  H  — p.  In  other  words,  they  can  occur  only 
where  the  corresponding  material  relation  is  already  present  in  the  system. 
Hence  for  every  theorem  in  the  system  in  which  a  relation  of  the  type 
poq  is  asserted,  there »is  a  theorem  in  which  the  corresponding  material 
relation,  p  q,  is  asserted. 

Consequently,  for  every  theorem  in  the  system  in  which  the  main  relation 
is  strict  there  is  an  exactly  similar  theorem  in  which  the  main  relation  is  the 
corresponding  material  relation. 

Wherever  strict  relations  appear  as  subordinate,  or  unasserted,  relations 
in  theorems,  the  situation  is  quite  similar.  These%  are  reducible  to  the 


310  A  Survey  of  Symbolic  Logic 

corresponding  material  relations  through  some  use  of  1-8  and  its  conse- 
quences. Note  particularly  theorem  2-77, 

(p  4  g)  =  (-  ~p  -{  -  ~g)  =  (~  -p  H  ~  -g)  =  (-q  -{  -p)  =  (~q  H  ~p) 

The  truth-value  of  any  strict  relation  will,  by  its  definition,  be  [~]  or  [~  -] 
or  [-  ~].  And  where  two  such  are  connected  by  H  or  any  equivalent  rela- 
tion, they  may  be  replaced  by  the  corresponding  relation  whose  truth-value 
is  simply  positive  or  is  [-] — and  this  is  always  a  material  relation. 

We  may  now  illustrate  this  reduction  of  subordinate  strict  relations: 
4-  3     [(p  •*  g)  -J  (r  •*  s)]  H  [(p  c  g)  -1  (res)] 

2-7  {p  -q/p;  r  -s/q} :  [-(p  -q)  -J  ~(r  -s)]  H  [-(p  -g)  H  -(r  -*)]         (1) 

1  -02:  (1)  =  [(p  -J  q)  -J  (r  -J  «)]  H  [-(p  -g)  -J  -(r  -*)]  (2) 

1-03:  (2)  =  Q.E.D. 
4-J.f     [(pAg)-j(rA*)]-i[(p  +  g)-i(f  +  *)] 

Similar  proof,  (-p  -g)  in  place  of  (p  -g),  etc. 
4- 32     [(p  o  q)  H  (r  o  5)]  -J  [(p  g)  H  (r  s)] 

2-75  {pg/p;  r*/g}:  [- ~(p  g)  H  -  ~(r  5)] -J  [(p  g) -J  (r  5)]  (1) 

1-01:  (1)  =  Q.E.D. 
4-  33     [(p  •*  g)  -*  (r  •*  5)]  -I  [(p  c  g)  c  (re*)] 

4-15:  [(pcg)-i(rc5)]-l[(pcg)c(rc5)]  (1) 

1-6:  4-3  x(l)i  Q.E.D. 
-#-J4     [(p  Ag)  -j  (r  AS)]  -1  [(p  +  g)  c  (r  +  *)] 

Similar  proof,  using  4-31  in  place  of  4-3. 
4- 35     [(p  o  g)  -}  (r  05)]  -J  [(p  g)  c  (r  5)] 

Similar  proof,  using  4-32. 
Note  that  as  a  subordinate  relation,  p  o  q  reduces  directly. 

In  theorems  4  •  3-4  •  32,  postulate  1  •  8  only  has  been  used,  and  the  reduc- 
tion of  strict  relations  to  material  relations  is  incomplete.  In  theorems 
4  •  33-4  •  35,  postulates  1  •  8  and  1  •  7  have  both  been  used,  and  the  reduction 
is  complete.  In  these  theorems,  dyads  of  dyads  are  dealt  with.  The 
reduction  extends  to  dyads  of  dyads  of  dyads,  and  so  on.  We  may  illustrate 
this  by  a  single  example  which  is  typical. 

Hypothesis:  [(p  •*  g)  -4  (-p  A  g)]  H  [(p  o  -g)  -J  -(p  -J  g)] 
To  prove:  [(p  c  g)  c  (-p  +  g)]  c  [(p  -g)  c  -(p  c  g)] 

(The  hypothesis  is  true,  though  it  has  not  been  proved.) 

2-71  {p  -q/p',  p  -q/q] :  [-(p  -q)  -J  -(p  -g)]  -J  [~(P  -q)  •»  ~(P  -?)]     (1) 
1-02,  1-03,  1-04,  1-05,  and  2-51: 

(1)  =  ((p  c  g)  -J  (-p  +  g)]  -J  [(p  -J  g)  -J  (-p  A  g)]     (2) 


The  System  of  Strict  Implication  311 

1-6:  (2)*Hyp.-l[(pcq)4(-p+q)]*[(po-q)4-(piq)]  (3) 

2-72  {p  -q/p;  p  -q/q} :  [-  ~(p  -g)  -J  -  ~(p  -g)]  -{ [(p  -9)  -j  (p  -g)]  (4) 
1-01,  1-02,  1-03,  and  2-51: 

(4)  =  [(p  o  -g)  -l  -(p  -J  g)]  -J  [(p  -g)  •*  -(p  c  g)]  (5) 

1-6:   (3)x(5)-i[(pcg)-J(-p+g)]-J[(p-g)H-(pcg)]  (6) 
4-33:  (6)-J  Q.E.D. 

In  any  theorem  in  which  ~p  is  related  to  ~g,  or  -  ~p  to  -  ~g,  or  ~  -p  to 
~  -g,  ~  may  be  replaced  by  -.     This  follows  immediately  from  2-77.     We 
illustrate  briefly  the  reduction  in  those  cases  in  which  ~r,  or  -  ~r,  or  ~  -r, 
is  related  to  p  o  g,  or  p  A  g,  or  p  -J  g. 
4-J6     (p  og-j-  ~r)  -{ (p  g-J  r) 

2-75  fp  g/p;  r/g} :  [-  -(p  g)  H  ~r]  -{  (p  g  -J  r)  (1) 

1-01:  (1)  =  (pogn-~r)H(pg-*r)  (2) 

4-15:  (pq-ir)  4  (pqcr)  (3) 

1-6:  (2)  x  (3) -i  Q.E.D. 
4-J7     (p  Ag-{  ~r)  -J  (p+  gc-r) 

2-7  {-p  -g/p;  r/g} :  [~(-p  -g)  -J  ~r]  -l  [-(-p  -g)  -J  -r]  (1) 

4.15:  [-(-p  -g)  -J  -r]  -I  [-(-p  -g)  c  -r]  (2) 

1-6:  (l)x(2)-j[~(-p-g)-i~rH[-(-p-g)c-r]  (3) 

1-04  and  1-05:  (3)  =  Q.E.D. 

A  dyad  of  triadic  strict  relations,  e.  g.,  p  o  (g  r)  -l  g  o  (p  r),  reduces 
just  like  a  dyad  of  dyads,  because  a  triadic  strict  relation  is  a  dyadic  strict 
relation — with  a  dyadic  material  relation  for  one  member.  But  a  triad 
of  dyadic  strict  relations  behaves  quite  differently.  Such  is  postulate  1  •  6, 

(p-Jg)(g-*r)-{(p-lr) 
This  does  not  look  like  a  strict  triad,  but  it  is,  being  equivalent  to 

(p  -I  g)  -J  [(g  -J  r)  c  (p  ^  r)] 

which  obviously  has  the  character  of  strict  triads  generally.  The  sub- 
ordinate relations  in  such  a  triad  cannot  be  reduced  by  any  direct  use  of 
1-8  and  its  consequences.  However,  all  such  relations  can  be  reduced. 
The  method  will  be  illustrated  shortly  by  deriving 

(pcg)(gcr)c(pcr) 
from  the  above. 

What  strict  relations,  then,  cannot  be  reduced  to  the  corresponding 
material  relations?  The  case  of  asserted  relations  has  already  been  dis- 


312  A  Survey  of  Symbolic  Logic 

cussed.  For  subordinate  relations,  the  question  admits  of  a  surprisingly 
simple  answer.  All  the  relations  of  the  system  can  be  expressed  in  terms 
of  some  product  and  the  various  truth  values — the  truth  values  of  ~  -p,  p, 
-  ~p,  -p,  and  ~p.  Let  us  remind  ourselves: 

poq  =  -~(pq)  p  q  =  -  -(p  q) 

p  4  q  =  ~(p  -q)  p  c  q  =  -(p  -q) 

p  A  q  =  ~(-p  -q)  p  +  q  =  -(-p  -q) 

The  difference  between  the  truth-value  of  p  and  that  of  -p,  between  ~p 
and  ~  -p,  between  -  ~p  and  -  ~  -p,  does  not  affect  reduction,  because  ~  -p 
can  be  regarded  as  ~  -(p)  or  as  ~(-p);  -  ~  -p  as  -  ~  -(p}  or  as  -  ~(-p); 
and  p  is  also  -(-p).  Hence  we  may  group  the  various  types  of  expression 
which  can  appear  in  the  system  under  three  heads,  according  to  truth- value: 

H  or  [~  -]  [  ]  or  [-]  [-  ~]  or  [-  -  -] 

p*q  pcq 

p  =  q  p  =  q 

p*q  p+q 

pq  poq 

.    -(p  o  9)  -(P  q) 

-(p  +  q)  -(p*q) 

-(p  =  q)  -(p  =  q) 

-(pcq)  ~(p-*q) 

~p  -p  p 

~  -p  p  — p 

In  this  table,  the  letters  are  quite  indifferent:  replacing  either  letter  by 
any  other  letter,  or  by  a  negative,  or  by  any  relation,  throughout  the  table, 
gives  a  valid  result.  The  blank  spaces  in  the  table  could  also  be  filled; 
for  example,  the  first  line  in  the  third  column  would  be  -  ~  -(p  -q).  But, 
as  the  example  indicates,  the  missing  expressions  are  more  complex  than 
any  which  are  given,  and  possess  little  interest.  The  significance  of  the 
table  is  this:  //,  in  any  theorem,  two  expressions  which  belong  in  the  same 
column  of  this  table  are  connected,  then  these  expressions  may  be  reduced  by 
postulate  1-8  and  its  consequences.  For,  by  2-77,  a  relation  of  any  two  in 
the  same  column  gives  the  corresponding  relation  of  the  corresponding  two 
in  either  of  the  other  columns.  But  any  theorem  which  relates  expressions 
which  belong  in  different  columns  of  this  table  is  not  thus  reducible,  since 
any  such  difference  of  truth-value  is  ineradicable.  This  table  also  sum- 


The  System  of  Strict  Implication  313 

marizes  the  consequences  of  postulate  1-7:  any  expression  in  the  table 
gives  the  expression  on  the  same  line  with  it  and  in  the  next  column  to  the 
right.  It  follows  that  expressions  in  the  column  to  the  left  also  give  the 
expressions  on  the  same  line  in  the  column  to  the  right,  since  -J  is  transitive. 
Just  as  postulate  1-7  is  the  only  source  of  asserted  strict  relations 
which  are  not  replaceable  by  the  corresponding  material  relations,  so  also 
the  only  theorems  containing  irreducible  subordinate  relations  are  con- 
sequences of  1-7.  For  this  postulate  is  the  only  one  in  which  different 
truth-values  are  related,  and  is  the  only  assumed  principle  by  which  an 
asserted  (or  denied)  truth-value  can  be  altered.  But  if  we  simply  substi- 
tute -  for  ~  in  1  •  7,  it  becomes  the  truism,  -p  -*  -p.  As  a  consequence,  for 
every  proposition  in  the  system  which  contains  strict  relations  or  the  truth- 
values,  [~],  [~  -],  [-  ~],  or  [- — ],  in  any  form,  in  such  wise  that  these  truth- 
values  cannot  be  reduced  to  the  simple  negative,  [-],  or  the  simple  positive 
(the  truth- value  of  p),  by  the  use  1-8,  the  theorem  which  results  if  we 
simply  substitute  -  for  ~  in  that  proposition  is  a  valid  theorem.  Or,  to 
put  it  more  clearly,  if  less  accurately;  if  any  theorem  involve  [~],  explicitly 
or  implicitly,  in  such  wise  that  it  cannot  be  reduced  to  [-]  by  the  use  of  1-8, 
still  the  result  of  substituting  ~  for  -  is  valid.  For  example,  4  •  13,  ~  -p  -i  —  p, 
cannot  be  reduced  by  1  •  8 ;  — p  and  -  ~p  are  irreducibly  different  truth- 
values.  But  substituting  -  for  ~,  we  have  -(-p}  -J  -(-p),  and  hence  -(-p) 
c-(-p),  or  pep.  Propositions  such  as  the  pair  ~  -p  -J — p  and  -(-p) 
c-(-p)  may  be  called  "pseudo-analogues".  If  we  reduce  completely,  so 
far  as  possible,  all  the  propositions  which  involve  [~]  or  strict  relations,  by 
the  use  of  1  •  7  and  1  •  8  and  their  consequences,  and  then  take  the  pseudo- 
analogues  of  the  remaining  propositions,  we  shall  find  such  pseudo-analogues 
redundant.  They  will  all  of  them  already  be  present  as  true  analogues  of 
propositions  which  are  completely  reducible.  This  transformation  by 
means  of  postulates  1  •  7  and  1  •  8,  by  which  strict  relations  give  the  cor- 
responding material  relations,  may  be  represented  by  the  substitution 
scheme 

peg,  p  =  q,  -(pq),      p+q,  -p,      p,         -(p) 
p  -I  q,  p  =  q,  -(p  O  g),  p  A  q,  ~p,  ~  -p,  -(-  ~p) 

We  put  -(p  o  q)  and  -(p  q),  -(-  ~p)  and  -(p),  because  p  o  q  as  a  main  rela- 
tion in  theorems  is  reducible  only  when  it  is  denied,  and  -  ~p  is  reducible 
only  through  its  negative.  As  we  have  now  shown  (except  for  triads  of 
dyads,  the  reduction  of  which  is  still  to  be  illustrated),  propositions  involving 


314 


A  Survey  of  Symbolic  Logic 


expressions  below  the  line  are  still  valid  when  the  corresponding  expressions 
above  the  line  are  substituted. 

The  transformation  by  {-/~\  of  all  the  assumptions  and  theorems  of  the 
system  of  Strict  Implication  which  can  be  thus  completely  reduced,  and  the 
rejection  of  remaining  propositions  which  involve  expressions  below  the  line 
(or  the  substitution  for  them  of  their  pseudo-analogues},  gives  precisely  the 
system  of  Material  Implication. 

All  the  postulates  and  theorems  of  Material  Implication  can  be  derived 
from  the  postulates  and  definitions  of  Strict  Implication:  the  system  of 
Strict  Implication  contains  the  system  of  Material  Implication.  We  may 
further  illustrate  this  fact  by  deriving  from  previous  propositions  the 
postulates  and  definitions  of  the  calculus  of  elementary  propositions  as 
it  appears  in  Principia  Mathematical 


4  •  41 


4  •  42 


4  •  4  3 


p  c  q  =  -p  +  q 

is  theorem  4-23. 

p  q  =  -(-p  +  -q) 

1-05  {-p/p;  -q/q}:  -p  +  -q  =  -[-(-p)  -(-?)] 

2-51:  (1)  =-p  +  -q  =  -(pq-) 

2-63:  (2)  =-(-p  +  -g)  =-(-(pq-)} 

2-51:  (3)  =  Q.E.D. 
(p  =  q)  =  (pcq)(qcp) 

is  the  definition,  1-07. 
p  +  pcp 

3-23:  p  +  p-lp 


(Principia,  *1-01) 

(Principia,  #3-01) 

(1) 
(2) 
(3) 

(Principia,  *4-01) 
(Principia,  #1-2) 


4 

•15: 

(1) 

H  Q.E.D. 

9<= 

P 

+  ? 

1 

•2   { 

-q/p 

;  -p/q}-  -p-q-t-q 

2 

•61: 

(1) 

•iq-i  -(-p  -q) 

1 

•05: 

(2) 

=  q  -Jp  +  q 

4 

•15: 

(3) 

-J  Q.E.D. 

p  + 

4 

cq- 

*p 

3 

•21: 

p  + 

q-iq  +  p 

4 

•15: 

(1) 

H  Q.E.D. 

(Principia, 


4-44 


4-45     p  +  (q  +  r)  c  q  +  (p  +  r) 

3-24:  p  +  (q  +  r)  -J  q  +  (p  +  r) 
4-15:  (l)n  Q.E.D. 
8  Pp.  98-101,  114,  120. 


(Principia, 


(Principia, 


(1) 


(1) 
(2) 
(3) 


(1) 


(1) 


The  System  of  Strict  Implication  315 

For  the  proof  of  the  last  postulate  in  the  set  in  Principia  Mathematica 
certain  lemmas  are  needed  which  are  of  interest  on  their  own  account. 

4-51     p  qcr  =  pc(qcr)  =  qc(pcr) 

1-03   {pqfp;  r/q}:  pqcr  = -[(pq)-r]  (1) 

2-91  and  2 -9:  (1)  =  pqcr  =  -[p(q-r}\  =  -[q(p -r)]  (2) 

2-51:  (2)  =  pqcr  =  -{p-[-(q-r-)}}  =  -{q  -[-(p  -r)]}  (3) 

.       1-03:  (3)  =  Q.E.D. 

4-52    p  q-tr  =  p*  (qcr)  =  q-i  (per) 

1-02   {p  q/p',  r/q} :  pq-ir  =  ~[(p  q)  -r] 
Remainder  of  proof,  similar  to  the  above. 

4-  S3     [(pcq)p]4q 

2-4   {pcq/p}:  (peg) -{(peg)  (1) 

4-52   {pcq/p;  p/q;  q/r}:  (1)  =  Q.E.D. 

It  is  an  immediate  consequence  of  4-53  that  "  If  p  is  asserted  and  p  cq 
is  asserted,  then  q  may  be  asserted",  for,  by  our  assumptions,  if  p  is  asserted 
and  p  c  q  is  asserted,  then  [(p  c  q)p]  may  be  asserted.  And  if  this  is  asserted, 
then  by  4-53  and  our  operation  of  "inference",  q  can  be  asserted.  But 
note  that  the  relation  which  validates  the  assertion  of  q  is  the  relation  -{  in 
the  theorem.  This  principle,  deduced  from  4-53,  is  required  in  the  system 
of  Material  Implication  (see  Principia,  *!•!  and  *1-11). 

4-54     (~pc~g)-i(gcp) 

4-3:  2-21 -j  Q.E.D. 

4-55     (p  -J  g)  -J (p  r  c  q  r) 

1  •  6  i-r/r } :  (p  4  g)  (q  -I  -r)  -J  (p  -J  -r)  (1) 

4-52:  (1)  =  (p  -5  g)  -J  [(q  -l  -r)  c  (p  -J  -r)}  (2) 

1-02  and  2-51:  (2)  =  (p  -{  g)  -i  [~(g  r)  c  ~(p  r}}  (3) 

4-54:  ~(gr)  c~(pr) -{ (p  r  cgr)  (4) 

1-6:  (3)  x  (4) -J  Q.E.D. 

4  •  56     (p  c  g)  c  (p  r  c  q  r) 

4-55  {(p  cq)p/p}:  4-53  •*  [(p  cg)p]r  cgr  (1) 

2-91:  (1)  =  [(pcq)(pr)]cqr  (2) 

4-51:  (2)  =  Q.E.D. 

4 •  57     p  cq  =  -q  c-p 

4-3:  2-62  -i  (peg)  -i(-gc-p)  (1) 

4-3:  2-6-J(-gc-p)-J(pcg)  (2) 

1-06:  (1)  x(2)  =  Q.E.D. 


316  A  Survey  of  Symbolic  Logic 

4-58     (pcq)(qcr)c(pcr) 

4-56  {-r/r}:  (p  cq)  c  (p-r  cq-r)  (1) 

4-57:  (1)  =  (peg)  c[-fo-r)  c-(p-r)]  (2) 

1-03:  (2)  =  (peg)  c  [foe  r)c  (per)]  (3) 

4-51:  (3)  =  Q.E.I5. 

4  •  58  is  the  analogue,  in  terms  of  material  relations,  of  1  •  6.  The  method 
by  which  we  pass  from  1-6  to  4-58  illustrates  the  reduction  of  triads  of 
strict  dyads  in  general.  This  reduction  begins  in  the  first  line  of  the  proof 
of  4-55.  Here  1  •  6  is  put  in  the  form 

(P<*q)4[(q*r)c(p-ir)] 
12345 

The  relation  numbered  4  is  already  a  material  relation.  This  is  character- 
istic of  strict  triads.  Relations  3  and  5  are  reduced  together  by  some 
consequence  of  1-8,  in  a  form  in  which  the  asserted  relation  is  material. 
Then,  as  in  4  •  56,  relations  1  and  2  are  reduced  together  by  the  use  of  4  •  53 
as  a  premise.  This  use  of  4-53  is  quite  puzzling  at  first,  but  will  become 
clearer  if  we  remember  its  consequence,  "  If  p  is  asserted  and  p  c  q  is  asserted, 
then  q  may  be  asserted".  This  method,  or  some  obvious  modification  of  it, 
applies  to  the  reduction  of  any  triad  of  strict  dyads  which  the  system  gives. 
We  can  now  prove  the  last  postulate  for  Material  Implication. 

4  •  59     (q  c  r)  c  [(p  +  9)  c  (p  +  r}]  (Principia,  *1  •  6) 

4-58   {-pip}:  (-pcg)focr)c(-pcr)  (1) 

4-51:  (1)  =  (gcr)c[(-pc(7)c(-pcr)]  (2) 
4-24:  (2)  =  Q.E.D. 

These  are  a  sufficient  set  of  symbolic  postulates  for  Material  Impli- 
cation, as  the  development  of  that  system  from  them,  in  Principia  Mathe- 
matica,  demonstrates.  However,  in  the  system  of  Strict  Implication,  those 
theorems  which  belong  also  to  Material  Implication  are  not  necessarily 
derived  from  the  above  set  of  postulates.  They  can  be  so  derived,  but  the 
transformation  {-/*}  produces  them,  more  simply  and  directly,  from  their 
analogues  in  terms  of  strict  relations. 

IV.    EXTENSIONS  OF  STRICT  IMPLICATION.    THE  CALCULUS  OF  CONSIST- 
ENCIES AND  THE  CALCULUS  OF  ORDINARY  INFERENCE 

From  the  symmetrical  character  of  postulate  1-8,  and  from  the  fact 
that  postulate  1-7  is  converted  by  negating  both  members,  i.  e.,  p-i-  ~p, 


The  System  of  Strict  Implication  317 

it  follows  that,  since  the  transformation  {-/-}  is  possible,  an  opposite 
transformation,  {~/-},  is  possible.  And  since  implications  are  reversed 
by  negating  both  members,  those  expressions  which  are  transformed  directly 
by  {-/~}  will  be  transformed  through  their  negatives  by  {~/-|,  while  those 
expressions  which  are  transformed  through  their  negatives  by  {-/~}  will 
be  transformed  directly  by  { ~/- } .  Hence  wre  have 


•(pcq),  -(p  =  q),  pq,  -(p  +  q)  -(-p),    -(p),      p 

This  substitution  scheme  may  be  verified  by  reference  to  the  table  on 
page  312.  The  transformation  {-/~}  represents  the  fact  that  expressions 
in  the  column  to  the  left,  in  this  table,  give  expressions  in  the  middle  column : 
{~/-}  represents  the  fact  that  expressions  in  the  middle  column  give  ex- 
pressions in  the  column  to  the  right.  {-/~}  eliminated  strict  relations: 
{*/-}  eliminates  material  relations.  As  in  the  previous  case,  so  here,  a 
dyad  of  dyadic  relations,  or  a  relation  connected  with  p,  -p,  ~p,  etc.,  can 
be  transformed  by  1-8  and  its  consequences  when  and  only  when  the 
connected  expressions  appear  in  the  same  column  of  that  table.  Thus  the 
transformation  {~/-}  is  subject  to  the  same  sort  of  limitation  as  is  {-/-}. 

The  transformation  {~/-},  eliminating  material  relations,  has  already 
been  illustrated  in  those  tables  in  Section  II,  in  which  theorems  in  terms  of 
strict  relations  were  compared  with  analogous  propositions  in  terms  of 
material  relations.  Theorems  in  the  right-hand  column  of  those  tables 
result  from  those  in  the  left-hand  column  by  the  transformation  {-/-}. 
The  proofs  of  3-11,  3-12,  3-13,  3-14,  3-31,  3-32,  3-33,  3-34,  3-52,  3-53, 
and  3  •  55  indicate  the  method  of  this  transformation.  Theorem  3  •  54  indi- 
cates a  limitation  of  it.  As  we  have  noted,  triadic  strict  relations  are  not 
expressible  in  terms  of  strict  dyads  alone.  Consequently,  in  the  case  of 
triadic  relations,  the  transformation  {~/-}  cannot  be  completely  carried 
out.  This  is  an  important  limitation,  since  postulate  1  •  6,  which  is  necessary 
for  any  generality  of  proof,  is  a  triadic  strict  relation.  It  means  that  any 
system  of  logic  in  which  there  are  no  material  relations  cannot  symbolize 
its  own  operations.  Since  strict  relations  are  the  relations  of  intension, 
this  is  an  important  observation  about  calculuses  of  intension  in  general. 

The  vertical  line  in  the  substitution  scheme  is  to  indicate  that  the 
transformation  {~/-}  is  arbitrarily  considered  to  be  complete  when  no 
material  relations  remain  in  the  expression,  p  and  -p  will  be  transformed 
when  connected  with  a  material  relation  which  is  transformed;  when  not 


318  A  Survey  of  Symbolic  Logic 

so  connected,  p  and  -p  remain.    They  could  be  transformed  in  all  cases, 
but  the  result  is  needlessly  complex  and  not  instructive. 

The  system,  or  partial-system,  which  results  from  the  transformation 
{~/-}  may  be  called  the  Calculus  of  Consistencies.  It  can  be  generated 
independently  by  the  following  assumptions: 

Let  the  primitive  ideas  be:  (1)  propositions,  p,  q,  r,  etc.,  (2)  -p,  (3)  ~p, 
(4)  p  o  q,  (5)  p  =  q. 

Let  the  other  strict  relations  be  defined  : 


II.     p  •*  q  =  -(p  O  -g) 

For  postulates  assume  : 
III.     p  o  q  4  q  o  p 
IV.    q  o  p  4  -  "p 
V.    -  ~p  -I  p  o  p 
VI.    p  =  -(-p) 

Assume  the  operations  of  "Substitution"  and  "Inference"  as  before, 
but  in  place  of  "Production"  put  the  following:  If  p  -J  q  is  asserted  and  q  -t  r 
is  asserted,  then  p  -*  r  may  be  asserted.  By  this  principle,  proof  is  possible 
without  the  introduction  into  the  postulates  of  triadic  relations. 

The  system  generated  by  these  assumptions  is  purely  a  calculus  of 
intensions.  It  is  the  same  which  would  result  from  performing  the  trans- 
formation {~/-}  upon  all  the  propositions  of  Strict  Implication  which 
admit  of  it,  and  rejecting  any  which  still  contain  expressions,  other  than  p 
and  -p,  below  the  line.  It  contains,  amongst  others,  all  those  theorems 
concerning  strict  relations  (except  the  triadic  ones)  which  were  exhibited 
in  Section  II  in  comparison  with  analogous  propositions  concerning  material 
relations. 

More  interest  attaches  to  another  partial-system  contained  in  Strict 
Implication.  If  our  aim  be  to  create  a  workable  calculus  of  deductive 
inference,  we  shall  need  to  retain  the  relation  of  the  logical  product,  p  q, 
but  material  implication,  p  c  q,  and  probably  also  the  material  sum,  p  •*•  q, 
may  be  rejected  as  not  sufficiently  useful  to  be  worth  complicating  the 
system  with.  The  ideas  of  possibility  and  impossibility  also  are  unnecessary 
complications.  Such  a  system  may  be  called  the  Calculus  of  Ordinary 
Inference.  The  following  assumptions  are  sufficient  for  it. 


The  System  of  Strict  Implication  319 

Primitive  Ideas:   (1)  Propositions,  p,  q,  r,  etc.,  (2)  -p,  (3)  p  -J  q,  (4)  p  q, 

(5)  p  =  q. 

Definitions: 

A.  p  A  q  =  -p  -j  q 

B.  p  O  g  =  -(p  -{ -g) 

C.  (p  =  q)  =  (2Mg)(g-*p) 

[D.  p+  q  =  -(-p -q)]     Optional. 

Postulates: 

E.  (-p  •*  g)  -J  (-g  •*  p) 

F.  p  qlp 

G.  p  -ip  p 

H.    p(g  r)  -J  g(p  r) 
I.       p  H  -(-p) 

J.      (p-lg)(?-5r)-i(p-{r) 

K.    p  q-lp  oq 

L.     (p  q-ir  s)  =  (p  oq-tr  os) 

All  of  these  assumptions  are  propositions  of  the  system  of  Strict  Impli- 
cation. A.  is  4-22,  B.  is  4-26,  C.  is  1-06,  and  D.  is  1-05;  E.  is  2-3,  F.  is 
2-1,  G.  is  1-3,  H.  is  1-4,  I.  is  1-5,  J.  is  1-06,  K.  is  4-26,  and  L.  is  an  im- 
mediate consequence  of  4  •  32  and  4  •  35.  The  Calculus  of  Ordinary  Inference 
is,  then,  contained  in  the  system  of  Strict  Implication.  It  consists  of  all 
those  propositions  of  Strict  Implication  which  do  not  involve  the  relation 
of  material  implication,  peg  [or  the  material  logical  sum,  p  +  q].  But 
where,  in  Strict  Implication,  we  have  ~p,  we  shall  have,  in  the  Calculus  of 
Ordinary  Inference,  -(pop}  or  p-t-p.  Similarly  — p  will  be  replaced 
by  ~(~P  o-p)  or  -p  -i  p,  and  — p  by  p  op  or  -(p  H  -p).  In  other  words, 
for  'p  is  impossible'  we  shall  have  'p  is  not  self-consistent'  or  'p  implies 
its  own  negation';  for  'p  is  necessary'  we  shall  have  'the  negation  of  p  is 
not  self-consistent '  or  '  the  negation  of  p  implies  p ' ;  and  for  '  p  is  possible ' 
we  shall  have  'p  is  self -consistent '  or  'p  does  not  imply  its  own  negation'. 

The  Calculus  of  Ordinary  Inference  contains  the  analogues,  in  terms  of 
p  q,  p  A  q,  and  p  H  q,  of  all  those  theorems  of  Material  Implication  which 
are  applicable  to  deductive  inference.  It  does  not  contain  the  useless  and 
doubtful  theorems  such  as  "A  false  proposition  implies  any  proposition", 
and  "A  true  proposition  is  implied  by  any  proposition".  As  a  working 


320  A  Survey  of  Symbolic  Logic 

system  of  symbolic  logic,  it  is  superior  to  Material  Implication  in  this 
respect,  and  also  in  that  it  contains  the  useful  relation  of  consistency,  p  o  q. 
On  the  other  hand,  it  avoids  that  complexity  which  may  be  considered  an 
objectionable  feature  of  Strict  Implication. 

The  system  of  Strict  Implication  admits  of  extension  to  propositional 
functions  by  methods  such  as  those  exhibited  in  the  last  chapter.  For 
the  working  out  of  this  extension,  several  modifications  of  this  method  are 
desirable,  but,  for  the  sake  of  brevity,  we  shall  adhere  to  the  procedure 
which  is  already  familiar  so  far  as  possible.  In  view  of  this,  the  outline 
to  be  given  here  should  be  taken  as  indicative  of  the  general  method  and 
results  and  not  as  a  theoretically  adequate  account.  Since,  as  we  have 
demonstrated,  the  system  of  Material  Implication  is  contained  in  Strict 
Implication,  It  follows  that,  with  suitable  definitions  of  Hpx  and  H<px, 
the  whole  theory  of  propositional  functions,  as  previously  developed,  may 
be  derived  from  Strict  Implication.  TL<px  will  here  be  interpreted  more 
explicitly  than  before,  "  <px  is  true  in  all  (actual)  cases,"  or  "  <px  is  true  of 
every  x  which  'exists'".  And  2<px  will  mean  "  <px  is  true  in  some  (actual) 
case",  or  "There  'exists'  at  least  one  x  for  which  <px  is  true".  The  novelty 
of  the  calculus  of  propositional  functions,  as  derived  from  Strict  Implica- 
tion, will  come  from  the  presence,  in  that  system,  of  ~p,  -  ~p,  ~  -p,  and  the 
strict  relations.  We  might  expect  that  if  tpx  is  a  propositional  function, 
~<px  would  be  a  propositional  function.  But  such  is  not  the  case:  ~<px  is  a 
proposition.  For  example,  "It  is  impossible  that  'x  is  a  man  but  not 
mortal'"  is  a  proposition  although  it  contains  a  variable.  So  is  "Nothing 
can  be  both  A  and  not-A",  which  predicates  the  impossibility  of  "x  is  A 
and  x  is  not-A".  It  would  be  an  error  to  suppose  that  all  the  propositions 
which  contain  variables  are  such  because  they  involve  the  idea  of  impossi- 
bility, or  necessity,  but  the  most  notable  examples,  the  laws  of  mathe- 
matics, are  propositions,  and  not  propositional  functions,  for  precisely  this 
reason.  When  stated  in  the  accurate  hypothetical  form — i.  e.,  as  the 
implications  of  certain  assumptions — they  are  necessary  truths. 

Since  ~  <px  is  a  proposition,  —  <px,  —  <px,  and  all  the  strict  relations  of 
propositional  functions  will  be  propositions.  If  <px  and  ^x  are  propositional 
functions,  then 

<px  o  \l/x  is  the  proposition  -  ~(<px  x  \j/x); 
<px  A  \l/x  is  the  proposition  ~(-<px  x> 
(f>x  H  \f/x  is  the  proposition  ~(<px  x- 


The  System  of  Strict  Implication  321 

We  shall  have  the  law,  ~  <px  -J  II  -  tpx,  "If  tpx  is  impossible,  then  it  is 
false  in  all  cases".  Hence  also,  2<px  -i-  ~(px,  "If  <px  is  sometimes  true, 
then  <px  is  possible".  The  first  of  these  gives  us  one  most  important  con- 
sequence, 

\f/x)  -J  Hx(  <f>x  c  \l/x) 


"If  it  is  impossible  that  <px  be  true  and  \l/x  false,  then  in  no  (actual)  case 
is  <px  true  but  $x  false",  or  "If  <px  strictly  implies  $x,  then  <f>x  formally 
implies  fa".  This  connects  the  novel  theorems  of  this  theory  of  prepo- 
sitional functions  with  the  better  known  propositions  which  result  from 
the  extension  of  Material  Implication.  Similarly  we  shall  have 


(  <px  A  fa}  ^H.x(<f>x  +  fa] 
and         S.r(<p£  *  fa}  H  (<px  o  fa} 


If  we  use  z(<pz)  to  denote  the  class  determined  by  <pz,  that  is,  the  class 
of  all  re's  such  that  <px  is  true,  then  we  derive  the  logic  of  classes  from  this 
calculus  of  prepositional  functions,  by  the  same  general  type  of  procedure 
as  that  exhibited  in  Section  III  of  Chapter  IV.  If  we  let  a  =  z(<pz~), 
j8  =  ztyz),  the  definitions  of  this  calculus  will  be  as  follows: 

(xn  e  a)  =  <pxn 

"x  is  a  member  of  the  class  a,  determined  by  the  function  <pz"  means 
"  <pxn  is  true". 

(a-i  /8)  =  (tfos-t  \l/x) 

(aC|8)   =  Ux(<pXC^x) 

(a  =  ]8)  =  (<px  =  $x) 

(a  =  ]8)  =  Ux(<px  =  fx) 

-a  =  x(-<px),         or         -a  =  x-(x6  a) 

(a  x  /3)  =  x(<px  x  \l/x),         or          (a  x  /3)  =  #[(#  e  a)  x  (x  e  /3)] 

(a  +  |8)  =  f(  ^  +  #e),         or         (a  +  |8)  =  ^[(*  c  a)  +  (a;  e  |8)] 

1  =  &($x  *  far) 

0  =  -1 

a  c  /3  is  the  relation  "All  members  of  a  are  also  members  of  /3"  —  a  relation 
of  extension.  It  is  defined  by  "In  every  (actual)  case,  either  <px  is  false 
or  \l/x  is  true";  or  "There  is  no  (actual)  case  in  which  <px  is  true  and  $x 
false"  —  Tlx(<px  c\J/x)  =  Ux(-<px  +  ^x~)  =  n2  -(<px  x-^a;).  a  -J  /3  is  the  cor- 
responding relation  of  intension  :  it  is  defined  by  "  Necessarily  either  <px  is 

false  or  \f/x  is  true",  or  "It  is  impossible  that  <px  be  true  and  fa  false", 

22 


322  A  Survey  of  Symbolic  Logic 

that  is,  (<px4\l/x)  =  (-<£>£  A  ^.r)  =  ~(<f>x  x-^a;).  a  -J  /3  may  be  correctly 
interpreted  "The  class-concept  of  a,  that  is,  <f>,  contains  or  implies  the 
class-concept  of  /3,  that  is,  \l/".  That  this  should  be  true  may  not  be  at 
once  clear  to  the  reader,  but  it  will  become  so  if  he  study  the  properties  of 
a  •*  j8,  and  of  <px  H  \f/x,  in  this  system. 

Since  we  have  (<px  -J  \f/x)  -I  Hx(<px  c  $x} 

we  shall  have  also  («-{/8)-{(aCj8) 

If  the  class-concept  of  a  implies  the  class-concept  of  /3,  then  every  member 
of  a  will  be  also  a  member  of  /3.  The  intensional  relation,  -J  ,  implies  the 
extensional  relation,  c.  But  the  reverse  does  not  hold.  The  old  "law"  of 
formal  logic,  that  if  a  is  contained  in  /3  in  extension,  then  /3  is  contained  in  a 
in  intension,  and  vice  versa,  is  false.  The  connection  between  extension 
and  intension  is  by  no  means  so  simple  as  that. 

This  discrepancy  between  relations  in  extension  and  relations  in  inten- 
sion is  particularly  evident  in  cases  where  one  of  the  classes  in  question  is 
the  null-class,  0,  or  the  universe  of  discourse,  1.  As  was  pointed  out  in 
Chapter  IV,  we  shall  have  for  every  "individual",  x, 

x  e  1,         and         -(x  e  0) 

Also,  for  any  class,  a,  we  shall  have 

a  c  1,         and         0  c  a 

These  last  two  will  hold  because,  since  fa;  -J  fa;  is  always  true  when  significant, 
-(fa;  -J  fa-)  always  false,  we  shall  have,  for  any  function,  <px, 


x<px  c  £x 

and         nz[-(fa;  -J  fa;)  c  <px] 

We  shall  have  these  because  "A  false  proposition  materially  implies  any 
proposition  ",  and  "  A  true  proposition  is  materially  implied  by  any  propo- 
sition." But  since  it  does  not  hold  that  "A  false  proposition  strictly  implies 
any  proposition",  or  that  "A  true  proposition  is  strictly  implied  by  any 
proposition  ",  we  shall  not  have 

<px  -{  (fa;  -J  fa;) 
or        -(fa;  -J  fa;)  -J  <px 

And  consequently  we  shall  not  have 

a  H  1,         or         0  -*  a 
If  7  is  a  null  class,  we  shall  have  "  All  members  of  7  are  also  members  of  j3, 


The  System  of  Strict  Implication  323 

whatever  class  ft  may  be".  But  we  shall  not  have  "The  class-concept  of  7 
implies  the  class  concept  of  ft,  whatever  class  ft  may  be".  The  implications 
of  a  class-concept  are  not  affected  by  the  fact  that  the  class  has  no  members. 
The  relation,  a  =  ft,  is  material  or  extensional  equivalence,  "The 
classes  a  and  ft  consist  of  identical  members";  a  =  ft  is  strict  or  intensional 
equivalence,  "The  class-concept  of  a  is  equivalent  to  the  class-concept  of  ft". 
It  is  obvious  that 

(a  =  ft}  -*  (a  =  ft} 

but  that  the  reverse  does  not  hold.  The  relation  between  intensions  and 
extensions  is  unsymmetrical,  not  symmetrical  as  the  medieval  logicians 
would  have  it.  And,  from  the  point  of  view  of  deduction,  relations  of 
intension  are  more  powerful  than  relations  of  extension. 

a+ ft  and  «x/3  are  relations  of  extension — the  familiar  "logical  sum" 
and  "logical  product"  of  two  classes.  What  about  the  corresponding 
relations  of  intension?  This  most  important  thing  about  them — there  are 
none.  Consider  the  equivalences, 

(a  A  ft}  =  x(<px  A  $x)  =  x[(x  e  a)  A  (x  e  ft}] 
and         (ao  ft}  =  x(<px  o  $x)  =  x[(x  t  at)  o  (x  e  ft}] 

<px  o\f/x  is  a  proposition — the  proposition  -  ~(<px  x^x},  "It  is  possible  that 
<px  and  \f/x  both  be  true".  And  being  a  proposition,  either  it  is  true  of 
every  x  or  it  is  true  of  none.  So  that  a  o  ft,  so  defined,  would  be  either 
1  or  0.  Similarly  <px  A  \f/x  is  a  proposition,  either  true  of  every  x  or  false 
of  every  x;  and  a  A  ft  would  be  either  1  or  0.  Consequently,  a  A  ft  and 
a  o  ft  are  not  relations  of  a  and  (8  at  all.  The  product  and  sum  of  classes 
are  relations  of  extension,  for  which  no  analogous  relations  of  intension 
exist.  This  is  the  clue  to  the  failure  of  the  continental  successors  of  Leibniz. 
They  sought  a  calculus  of  classes  in  intension :  there  is  no  such  calculus,  unless 
it  be  confined  to  the  relations  a  -i  ft  and  a  =  ft.  Holland  really  came  in 
sight  of  this  fact  when  he  pointed  out  to  Lambert  the  difficulties  of  logical 
"multiplication"  and  logical  "division".7 

The  presentation  of  the  calculus  of  prepositional  functions  and  calculus 
of  classes  here  outlined, — and  of  the  similar  calculus  of  relations, — would 
involve  many  subtle  and  vexatious  problems.  But  we  have  thought  it 
worth  while  to  indicate  the  general  results  which  are  possible,  without  dis- 
cussing the  problems.  But  there  is  one  important  problem  which  involves 
the  whole  question  of  strict  implication,  material  implication,  and  formal 

7  See  above,  p.  35. 


324  A  Survey  of  Symbolic  Logic 

implication,  which  must  be  discussed — the  meaning  of  "implies".     This 
is  the  topic  of  the  next  section. 

V.    THE  MEANING  OF  "IMPLIES" 

It  is  impossible  to  escape  the  assumption  that  there  is  some  definite 
and  "proper"  meaning  of  "implies".  The  word  denotes  that  relation 
which  is  present  when  we  "validly"  pass  from  one  assertion,  or  set  of 
assertions,  to  another  assertion,  without  any  reference  to  additional  "evi- 
dence". If  a  system  of  symbolic  logic  is  to  be  applied  to  such  valid  infer- 
ence, the  meaning  of  "implies"  which  figures  in  it  must  be  such  a  "proper" 
meaning.  We  should  not  hastily  assume  that  there  is  only  one  such 
meaning,  but  we  necessarily  assert  that  there  is  at  least  one.  This  is  no 
more  than  to  say:  there  are  certain  ways  of  reasoning  that  are  correct  or 
valid,  as  opposed  to  certain  other  ways  which  are  incorrect  or  invalid. 

Current  pragmaticism  in  science,  and  the  passing  of  "self-evident 
axioms"  in  mathematics  tend  to  confuse  us  about  this  necessity.  Pure 
mathematics  is  no  longer  concerned  about  the  truth  either  of  postulates  or 
of  theorems,  and  definitions  are  always  arbitrary.  Why,  then,  may  not 
symbolic  logic  have  this  same  abstractness?  What  does  it  matter  whether 
the  meaning  of  "implies"  which  figures  in  such  a  system  be  "proper"  or 
not,  so  long  as  it  is  entirely  clear?  The  answer  is  that  a  system  of  symbolic 
logic  may  have  this  kind  of  abstractness,  as  will  be  demonstrated  in  the  next 
chapter.  But  it  cannot  be  a  criterion  of  valid  inference  unless  the  meaning, 
or  meanings,  of  "implies"  which  it  involves  are  "proper".  There  are 
two  methods  by  which  a  system  of  symbolic  logic  may  be  developed:  the 
non-logistic  method  exemplified  by  the  Boole-Schroder  Algebra  in  Chapter 
II,  or  the  logistic  method  exhibited  in  Principia  Mathematica  and  in  the 
development  of  Strict  Implication  in  this  chapter.  The  non-logistic  method 
takes  ordinary  logic  for  granted  in  order  to  state  its  proofs.  This  logic  which 
is  taken  for  granted  is  either  "proper"  or  the  proofs  are  invalid.  And  if 
the  logic  it  takes  for  granted  is  not  the  logic  it  develops,  then  we  have  a 
most  curious  situation.  A  symbolic  logic,  logistically  developed — i.  e., 
without  assuming  ordinary  logic  to  validate  its  proofs — is  peculiar  among 
mathematical  systems  in  that  its  postulates  and  theorems  have  a  double 
use.  They  are  used  not  only  as  premises  from  which  further  theorems  are 
deduced,  but  also  as  rules  of  inference  by  which  the  deductions  are  made. 
A  system  of  geometry,  for  example,  uses  its  postulates  as  premises  only; 
it  gets  its  rules  of  inference  from  logic.  Suppose  a  postulate  of  geometry 


The  System  of  Strict  Implication  325 

to  be  perfectly  acceptable  as  an  abstract  mathematical  assumption,  but 
false  of  "our  space".  Then  the  theorems  which  spring  from  this  assump- 
tion may  be  likewise  false  of  "our  space".  But  still  the  postulate  will 
truly  imply  these  theorems.  However,  if  a  postulate  of  symbolic  logic, 
used  as  a  rule  of  inference,  be  false,  then  not  only  will  some  of  the  theorems 
be  false,  but  some  of  the  theorems  will  be  invalidly  inferred.  The  use  of 
the  false  postulate  as  a  premise  will  introduce  false  theorems;  its  use  as  a 
rule  of  inference  will  produce  invalid  proofs.  "  Abstractness "  in  mathe- 
matics has  always  meant  neglecting  any  question  of  truth  or  falsity  in 
postulates  or  theorems;  the  peculiar  case  of  symbolic  logic  has  thus  far 
been  overlooked.  But  we  are  hardly  ready  to  speak  of  a  "good"  abstract 
mathematical  system  whose  proofs  are  arbitrarily  invalid.  Until  we  are, 
it  is  requisite  that  the  meaning  of  "implies"  in  any  system  of  symbolic 
logic  shall  be  a  "proper"  one,  and  that  the  theorems — -used  as  rules  of 
inference — shall  be  true  of  this  meaning. 

Unless  "implies"  has  some  "proper"  meaning,  there  is  no  criterion  of 
validity,  no  possibility  even  of  arguing  the  question  whether  there  is  one  or 
not.  And  yet  the  question  What  is  the  "proper"  meaning  of  "implies"? 
remains  peculiarly  difficult.  It  is  difficult,  first,  because  there  is  no  common 
agreement  which  is  sufficiently  self-conscious  to  decide,  for  example,  about 
"material  implication"  or  "strict  implication".  Even  those  who  feel  quite 
decided  in  the  matter  are  easily  confused  by  the  subtleties  of  the  problem. 
And,  second,  it  is  difficult  because  argument  on  the  topic  is  necessarily 
petitio  principii.  One  must  make  the  Socratic  presumption  that  one's 
interlocutor  already  knows  the  meaning  of  "implies",  and  agrees  with 
one's  self,  and  needs  only  to  be  made  aware  of  that  fact.  One  must  sup- 
pose that  the  meaning  in  denotation  is  clear  to  all,  as  the  meaning  of  "cat" 
or  "life"  is  clear,  though  the  definition  remains  to  be  determined.  If  two 
persons  should  really  disagree  about  "implies" — should  have  different 
"logical  sense" — there  would  be  nothing  to  hope  for  from  their  argument. 

In  consideration  of  this  peculiar  involution  of  logical  questions,  the 
best  procedure  is  to  exhibit  the  alternatives  in  some  detail.  When  the 
nature  of  each  meaning  of  "implies",  and  the  consequences  of  taking  it  to 
be  the  "proper"  one  have  been  exhibited,  the  case  rests. 

We  have  already  drawn  attention,  both  in  this  chapter  and  in  Chapter  II, 
to  the  peculiar  theorems  which  belong  to  all  systems  based  on  material 
implication.  We  may  repeat  here  a  few  of  them: 

(1)  A  false  proposition  implies  any  proposition;  -p  c  (p  c  q) 


326  A  Survey  of  Symbolic  Logic 

(2)  A  true  proposition  is  implied  by  any  proposition;  q  c  (p  cq) 

(3)  If  p  does  not  imply  q,  then  p  is  true;  -(p  cq)  cp 

(4)  If  p  does  not  imply  q,  then  q  is  false;  -(p  c  </)  c  -q 

(5)  If  p  does  not  imply  <?,  then  p  implies  that  q  is  false; 

-(pcg)c(pc-g) 

(6)  If  p  does  not  imply  9,  then  'p  is  false '  implies  q;  -(p  cq)  c  (-p  c  q) 

(7)  If  p  and  q  are  both  true,  then  p  implies  q  and  q  implies  p; 

pqc(pcq)(qcp) 

(8)  If  p  and  g  are  both  false,  then  p  implies  q  and  q  implies  p; 

-p-qc(pcq)(qcp) 

These  sufficiently  characterize  the  relation  of  material  implication.  It  is 
obviously  a  relation  between  the  truth-values  of  propositions,  not  between 
any  supposed  content  or  logical  import  of  propositions,  "p  materially 
implies  q"  means  "It  is  false  that  p  is  true  and  q  false".  All  these  the- 
orems, and  an  infinite  number  of  others  just  as  "peculiar"  follow  necessarily 
from  this  definition.  The  one  thing  which  this  relation  has  in  common 
with  other  meanings  of  "implies" — a  most  important  thing  of  course — is 
that  if  p  is  true  and  q  is  false,  then  p  does  not  materially  imply  q. 

As  has  been  said,  there  are  any  number  of  such  "peculiar"  theorems 
in  any  calculus  of  propositions  based  on  material  implication.8  These  the- 
orems do  not  admit  of  any  application  to  valid  inference.  In  a  system  of 
material  implication,  logistically  developed,  there  is  nothing  to  prohibit 
their  being  used  as  rules  of  inference,  but  when  so  used  they  give  theorems 
which  are  even  more  peculiar  and  quite  as  useless.  If  we  apply  these 
theorems  to  non-symbolic  propositions,  we  get  startling  results.  "The 
moon  is  made  of  green  cheese"  implies  "2  +  2  =  4", — because  q  c  (p  c  q). 
Let  q  be  "2  +  2  =  4"  and  p  be  "The  moon  is  made  of  green  cheese". 
Then,  since  "2  +  2  =  4"  is  true,  its  consequence  above  is  demonstrated. 
"If  the  puppy's  teeth  are  filled  with  zinc,  tomorrow  will  be  Sunday". 
Because  the  puppy's  teeth  are  not  filled  with  zinc,  and,  anyway,  it  happens 
to  be  Saturday  as  I  write.  A  false  proposition  implies  any,  and  a  true 
proposition  is  implied  by  any.9 

There  are,  then,  in  the  system  of  Material  Implication,  a  class  of  propo- 
sitions, which  do  not  admit  of  any  application  to  valid  inference.  And 

8  Every  theorem  gives  others  by  substitution,  as  well  as  by  being  used  as  a  rule  of 
inference.    And  there  are  ways  whereby,  for  any  such  theorem,  one  other  which  is  sure  to 
be  "peculiar"  also  can  be  derived  from  it.     And  also,  it  can  be  devised  so  that  no  result  of 
one  shall  be  the  chosen  result  of  any  other.     Hence  the  number  is  infinite. 

9  Lewis  Carroll  wrote  a  Symbolic  Logic.     I  shall  never  cease  to  regret  that  he  had  not 
heard  of  material  implication. 


The  System  of  Strict  Implication  327 

all  the  other,  non-"  peculiar ",  theorems  of  Material  Implication  find  their 
analogues  in  other  systems.  Hence  the  presence  of  these  peculiar  and  use- 
less theorems  is  a  distinguishing  mark  of  systems  based  upon  material 
implication. 

There  can  be  no  doubt  that  the  reason  why  the  relation  of  material 
implication  is  the  basis  of  every  calculus  of  propositions  except  MacColl's 
and  Strict  Implication  is  a  historical  one.  Boole  developed  his  algebra  for 
classes;  he  then  discovered  that  it  could  also  be  interpreted  so  as  to  cover 
certain  relations  of  propositions.  Peirce  modified  Boole's  algebra  by  intro- 
ducing the  relation  of  inclusion,  which  we  have  symbolized  by  c .  a  c  b 
has  all  the  properties  of  the  relation  between  a  and  b  when  every  member  of 
a  is  also  a  member  of  b.  It  has  one  notable  peculiarity:  if  a  is  a  class  which 
has  no  members — a  "zero"  class — then  for  any  class  x,  a  ex.  Now  the 
idea  of  "zero"  in  any  branch  of  mathematics  seems  a  little  more  of  an 
arbitrary  convention  than  the  other  numbers.  The  arithmetical  fact  that 
0  <  8  seems  "queer"  to  children,  and  it  would,  most  likely,  have  seemed 
"queer"  to  an  ancient  Roman.  Once  O.is  defined,  its  "queer"  properties, 
as  well  as  the  obvious  one,  8  +  0  =  8,  are  inevitable.  It  is  similar  with 
the  "null  class",  a  0  =  0,  "That  which  is  both  a  and  nothing  is  nothing", 
is  necessary.  And  (a  b  =  a)  =  (a  c6),  "'That  which  is  both  a  and  b, 
is  a'  is  equivalent  to  'All  a  is  &'",  leads  to  the  necessary  consequence 
Oca.  If  there  are  no  sea  serpents,  then  "All  sea  serpents  are  arthropods" 
necessarily  follows.  This  consequence  seems  more  "queer"  and  arbitrary 
because  it  is  a  relation  of  extension  with  no  analogue  in  intension.  The 
concept  "sea  serpent"  does  not  imply  the  concept  "arthropod" — as  has 
been  pointed  out,  0  -J  a  does  not  hold.  And  in  our  ordinary  logical  thinking 
we  pass  from  intension  to  extension  and  vice  versa  without  noting  the 
difference,  because  the  relations  of  the  two  are  so  generally  analogous. 
But  once  we  make  the  necessary  distinction  of  relations  in  extension  from 
relations  in  intension,  it  is  clear  that  0  c  a  in  extension  is  a  necessary  conse- 
quence of  the  concept  of  the  null-class.  Entirely  similar  remarks  apply  to 
the  proposition  a  c  1,  except  that  "a  is  contained  in  everything"  does  not 
seem  so  "queer". 

Boole  suggested  that  the  algebra  of  classes  be  reinterpreted  as  a  calculus 
of  propositions  by  letting  a,  b,  c,  etc.,  represent  the  times  when  the  proposi- 
tions A,  B,  C,  etc.,  are  true.  Then  Peirce  added  the  postulate  which 
holds  for  propositions  but  not  for  classes  (or  for  propositional  functions), 
a  =  (a  =  1).  A  proposition  is  either  true  in  all  cases,  or  true  in  none. 


328  A  Survey  of  Symbolic  Logic 

The  class  of  cases  in  which  any  proposition  is  true  is  either  0  or  1.  This 
gives  the  characteristic  property  of  the  Two-Valued  Algebra.  If  we  add 
to  this  the  interpretation  of  a  c  b,  "  All  cases  in  which  A  is  true  are  cases  in 
which  B  is  true",  or  loosely  "If  A,  then  B",  we  have  the  source  of  the 
peculiar  propositions  of  Material  Implication.  For  Oc&,  a  c  1,  OcO, 
0  c  1,  and  1  c  1  follow  from  the  laws  which  are  thus  extended  from  classes 
to  propositions.  A  false  proposition  [=  0]  implies  b.  And  a  implies  any 
true  proposition  [=  1].  Of  any  two  true  propositions  [=  1]  each  implies 
the  other.  Of  any  two  false  propositions  [=  0]  each  implies  the  other. 
And  any  false  proposition  implies  any  true  one.  "A  false  proposition 
materially  implies  any  proposition"  means  precisely  "If  there  are  no 
cases  in  which  A  is  true  (if  a  =  0)  then  all  cases  in  which  A  is  true  are 
also  cases  in  which  B  is  true".  It  does  not  mean  " B  can  be  inferred  from 
any  false  proposition".  "A  true  proposition  is  materfally  implied  by  any 
proposition"  means  only,  "If  B  is  true  [=  1],  then  the  cases  in  which  A  is 
true  are  contained  among  the  cases  (i.  e.,  all  cases)  in  which  B  is  true". 
It  does  not  mean  "Any  true  proposition  can  be  inferred  from  A  ".  Inference 
depends  upon  meaning,  logical  import,  intension,  a  c  b  is  a  relation  purely 
of  extension.  Is  this  material  implication,  a  c  b,  a  relation  which  can 
validly  represent  the  logical  nexus  of  proof  and  demonstration? 

Formal  implication  Ux(<pxc  \f/x)  is  defined  in  terms  of  material  implication. 
It  means  "For  every  value  of  x,  px  materially  implies  \f/x".  Choose  any 
value  of  x,  say  for  convenience  z,  and  unless  <pz  is  false,  $z  is  true.  Cer- 
tainly this  relation  approximates  more  closely  to  the  usual  meaning  of 
"implies".  But  the  precisely  accurate  interpretation  of  ILx(<px  c\{/x) 
depends  upon  what  is  meant  by  the  "values  of  x".  We  have  spoken  of 
them  as  "cases"  or  "individuals".  It  makes  a  distinct  difference  whether 
the  "cases"  comprehended  by  Iix(<px  c\l/x)  are  all  the  possible  cases,  all 
conceivable  individuals,  or  only  all  actual  cases,  all  individuals  which  exist 
(in  the  universe  of  discourse).  Either  interpretation  may  consistently  be 
chosen,  but  the  consequences  of  the  choice  are  important.  Let  us  survey 
briefly  the  more  significant  considerations  on  this  point. 

In  the  first  place,  supposing  that  the  second  choice  is  made  and  Uf  be 
taken  to  signify  "for  all  z's  which  exist",  what  shall  we  mean  by  "exist"? 
This  is  entirely  a  matter  of  convenience,  and  logicians  are  by  no  means  at 
one  in  their  use  of  the  term.  But  any  meaning  of  "exist"  which  confines 
it  to  temporal  and  physical  reality  or  to  what  is  sometimes  called  "the 
factual "  is  inconvenient  because,  for  example,  we  may  wish  to  distinguish 


The  System  of  Strict  Implication  329 

the  status  of  curves  without  tangents  in  mathematics  from  the  status  of 
the  square  of  the  circle.  This  distinction  is  usually  made  by  saying  that 
the  former  "exist",  since  the  general  mathematical  idea  of  a  "curve" 
admits  such  cases,  and  their  equations  may  be  given;  while  the  square  of 
the  circle  is  demonstrably  impossible.  Again,  it  is  inconvenient  to  say  that 
Apollo  exists  in  Greek  mythology,  whereas  the  god  Agni  does  not.  Now 
the  god  Agni  is  not  inconceivable  in  Greek  mythology;  we  find  no  record 
of  him,  that  is  all.  Similarly,  while  the  usual  illustrations  of  mathematical 
"non-existence"  are  impossibilities,  there  is  still  a  difference  between  what 
"does  not  exist"  in  a  mathematical  system  and  what  is  impossible.  Sup- 
pose we  have  an  "existence  postulate"  in  a  set  which  are  consistent  and 
independent  each  of  the  others — the  0-postulate  in  the  Boole-Schroder 
Algebra,  for  example.  Without  this  postulate,  the  remainder  of  the  set 
generate  a  system  in  which  0  does  not  exist.  But  it  is  possible,  as  the  con- 
sistency of  this  postulate  with  the  others  demonstrates.  The  frequent 
statement  that  "mathematical  existence"  is  the  same  as  "possibility"  is  a 
very  thoughtless  one. 

The  most  convenient  use  of  "exist"  in  logic  is,  then,  one  which  makes 
the  meaning  depend  upon  the  universe  of  discourse,  but  one  which  does 
not,  as  is  sometimes  supposed,  thereby  identify  the  "existent"  and  the 
possible.  ("Possible"  similarly  varies  its  meaning  with  the  universe  of 
discourse.)  On  the  Other  hand,  it  is  inconvenient  to  use  "exist"  so  widely 
that  "existence"  is  a  synonym  for  "conceivability".  This  is  so  obvious 
in  the  most  frequent  universe  of  discourse,  "phenomena",  that  it  hardly 
needs  to  be  pointed  out. 

Using  "exists"  in  this  sense,  in  which  "existence"  is  narrower  than 
"possibility"  but  may,  in  some  universes  of  discourse,  be  wider  than  "the 
factual",  it  makes  a  difference  whether  Hx  in  Ux(<pxcfa)  denotes  only 
existent  x's  or  all  possible  a;'s.  All  American  silver  coins  dated  1915  have 
milled  edges.  Let  <px  be  "x  is  an  American  silver  coin  dated  1915",  and 
let  fa  be  "x  has  a  milled  edge".  There  is  no  necessity  about  milled  edges 
for  silver  coins,  unless  one  speak  in  the  "legal"  universe  of  discourse.  For 
this  illustration,  Hx((f>xcfa)  will  be  true  if  Hx  denote  only  actual  x's; 
false  if  it  denote  all  possible  x's.  One  illustration  is  as  good  as  a  hundred; 
if  Hx(<px  c  fa)  refer  to  all  possible  x's,  Ux(<px  c  fa)  means  "It  is  impossible 
that  <px  be  true  and  fa  false".  If  Hx(<f>xcfa)  be  confined  to  actual  x's, 
then  it  signifies  a  relation  of  extension,  "The  class  of  things  of  which  ipx 
is  true  is  contained  in  the  class  of  things  of  which  fa  is  true". 


330  A  Survey  of  Symbolic  Logic 

It  might  be  thought  that  the  meaning  of  Hx(<px  c\l/x)  is  sufficiently 
determined  by  saying  that  the  "values  of  x"  in  a  function,  <px,  are  all  the 
entities  for  which  <px  is  either  true  or  false.  But  this  is  not  the  case,  for 
there  is  question  whether,  of  an  x  which  does  not  exist,  <px  is  always  true, 
or  always  false,  or  sometimes  true  and  sometimes  false,  or  never  either 
true  or  false.  Here  again,  the  question  is,  in  part,  one  of  convention. 
From  the  point  of  view  of  extension,  it  is  obvious  that  if  <?x  can  be  predi- 
cated at  all  of  an  x  which  does  not  exist,  it  will  always  be  false.  (Predicating 
something,  <p,  of  an  "individual",  x,  which  does  not  exist,  should  be  dis- 
tinguished from  asserting  that  an  empty  class,  a,  which  exists  though  it 
has  no  members,  is  included  in  some  other,  a  c  6.  "The  King  of  France  is 
bald"  is  an  example  of  the  former;  "All  sea  serpents  have  green  wings", 
of  the  latter.)  And  the  point  of  view  of  extension  is  frequently  that  of 
common  sense.  In  this  sense,  "a:  is  a  man"  is  false  of  my  non-existent  twin 
brother,  and  even  identical  propositions  such  as  "My  twin  brother  is  my 
twin  brother"  are  false  of  the  non-existent.  But  from  the  point  of  view  of 
intension,  an  identical  proposition  is  always  true,  and  <px  may  be  true  or 
it  may  be  false  of  a  non-existent  x.  If  the  point  of  view  of  extension  be 
taken  with  reference  to  prepositional  functions,  then  <px  is  either  not- 
significant  or  false  of  the  non-existent,  and  \f<x  is  similarly  not-significant 
or  false.  If  <px  and  \l/x  are  not  significant  of  the  non-existent,  then  Iix(<px 
c\j/x)  means  "For  every  existent  x,  px  materially  implies  \I/x".  If  <px 
and  $x  are  significant  and  false  of  the  non-existent,  then  <px  c  \f/x  is  true 
of  every  non-existent  x,  since  of  two  false  propositions,  each  materially 
implies  the  other.  Hence  on  this  interpretation,  Hx(<px  c\f/x~)  is  significant 
for  all  possible  x's  and  true  in  case  every  existent  x  is  such  that  <px  c  \j/x. 
Hence  its  meaning  will  still  be  accurately  rendered  by  "For  every  existent 
x,  <px  materially  implies  \l/x".  If  the  point  of  view  of  intension  be  taken 
with  reference  to  propositional  functions,  or  if  it  be  left  open,  then  H.x(<px 
c\f/x)  may  mean  "For  every  possible  x,  <px  materially  implies  $x",  or  we 
may,  by  convention,  still  confine  it  to  the  meaning  "For  every  existent  x, 
<px  materially  implies  i{/x".10 

10  We  would  gladly  have  spared  the  reader  these  details,  but  we  dared  not.  If  logicians 
do  not  consider  one  another's  views,  who  will?  In  this  connection  we  are  reminded  of  a 
passage  in  Lewis  Carroll's  Symbolic  Logic  (pp.  163-64)  anent  the  controversy  concerning 
the  existential  import  of  propositions: 

"The  writers,  and  editors,  of  the  Logical  text-books  which  run  in  the  ordinary 
grooves — to  whom  I  shall  hereafter  refer  by  the  (I  hope  inoffensive)  title  'The  Logi- 
cians'— take,  on  this  subject,  what  seems  to  me  to  be  a  more  humble  position  than  is 
at  all  necessary.  They  speak  of  the  Copula  of  a  Proposition  'with  bated  breath" 


The  System  of  Strict  Implication  331 

The  ground  being  now  somewhat  cleared,  we  return  to  the  simpler 
considerations  which  are  really  more  important.  What  are  the  conse- 
quences of  taking  TLx(<px  c\l/x)  in  one  or  the  other  meaning?  The  first 
and  most  important  is  this.  It  is  a  desideratum  that  we  should  be  able  to 
derive  the  calculus  of  classes  from  the  calculus  of  prepositional  functions. 
And  in  this  calculus  of  classes,  the  inclusion  relation  of  classes  a  c  @  or 
z(<pz)  c  ztyz)  can  be  defined  by 

z(if/Z)    =   Ux(<f>X  C  $X) 


or  by  some  equivalent  definition.  If  Ux(<px  c\f/x)  mean  "For  all  x's 
which  exist,  <pxc\f/x",  then  a  c  /3,  or  z(<pz)  cz(\f/z),  so  defined,  is  the  use- 
ful relation  of  extension,  "All  the  existing  things  which  are  members  of  a. 
are  also  members  of  /3".  Such  a  relation  can  represent  such  propositions  as 
"All  American  silver  coins  dated  1915  have  milled  edges".  If,  on  the  other 
hand,  we  interpret  Tix(<f>x  c\f/x)  to  mean  "For  all  possible  x's,  (pxc^f/x, 
then  two  courses  are  open:  (1)  we  can  maintain  that  whatever  is  true  of 
all  existent  things  is  true  of  all  possible  —  thus  abrogating  a  useful  and 
probably  indispensable  logical  distinction;  or  (2)  we  can  allow  that  what 
is  true  or  false  of  the  possible  depends  upon  its  nature  as  conceived  or 
defined.  If  we  make  the  second  choice  here,  the  consequence  is  that 
a  c  13,  or  z((pz)  c  z(\fsz),  defined  by 

z(<f>z)  c  z($z)  =  Hx(<px  c  \f/x) 
or  in  any  equivalent  fashion,  such  as 

a  c  j8  =  HI  (a:  e  a  ex  e  |8) 

is  the  relation  of  intension  "The  class-concept  of  a  implies  the  class- 
concept  of  /3".  This  relation  does  not  symbolize  such  propositions  as 

almost  as  if  it  were  a  living,  conscious  Entity,  capable  of  declaring  for  itself  what  it 
chose  to  mean  and  that  we,  poor  human  creatures,  had  nothing  to  do  but  to  ascertain 
what  was  its  sovereign  will  and  pleasure,  and  submit  to  it. 

"In  opposition  to  this  view,  I  maintain  that  any  writer  of  a  book  is  fully  authorised 
in  attaching  any  meaning  he  likes  to  any  word  or  phrase  he  intends  to  use.  If  I  find 
an  author  saying,  at  the  beginning  of  his  book,  'Let  it  be  understood  that  by  the 
word  ''black"  I  shall  always  mean  "white",  and  that  by  the  word  ''white"  I  shall 
always  mean  ''black",'  I  meekly  accept  his  ruling,  however  injudicious  I  may  think  it. 

"And  so,  with  regard  to  the  question  whether  a  Proposition  is  or  is  not  to  be 
understood  as  asserting  the  existence  of  its  Subject,  I  maintain  that  every  writer  may 
adopt  his  own  rule,  provided  of  course  that  it  is  consistent  with  itself  and  with  the 
accepted  facts  of  Logic. 

"Let  us  consider,  one  by  one,  the  various  views  that  may  logically  be  held,  and 
thus  settle  which  of  them  may  conveniently  be  held;  after  which  I  shall  hold  myself 
free  to  declare  which  of  them  7  intend  to  hold." 


332  A  Survey  of  Symbolic  Logic 

"All  American  silver  coins  dated  1915  have  milled  edges"  or  "It  rained 
every  week  in  March",  or  in  general,  the  frequent  universal  propositions 
which  predicate  this  relation  of  extension. 

And  whichever  interpretation  of  Ui(<fOfC\px)  be  chosen,  we  can  now 
point  out  one  interesting  peculiarity  of  it.  We  quote  from  Principia 
Mathematica:  u  "In  the  usual  instances  of  implication,  such  as  "'Socrates 
is  a  man'  implies  'Socrates  is  a  mortal'",  we  have  a  proposition  of  the 
form  "  <px  c  \f/x"  in  a  case  in  which  "nx(<px  c  \f/x) "  is  true.  In  such  a  case, 
we  feel  the  implication  as  a  particular  case  of  a  formal  implication  ".  It 
might  be  added  that  " '  Socrates  is  a  man '  implies  '  Socrates  is  a  mortal ' ' 
is  not  a  formal  implication:  it  is  a  material  implication  and  a  strict  impli- 
cation, but  not  formal.  One  may  object:  "But  as  a  fact,  in  such  cases 
there  is  a  tacit  premise  of  the  type  'All  men  are  mortal ',  and  this  is  precisely 
the  formal  implication,  Ux((px  c\f/x)".  Granted,  of  course.  But  add  this 
premise,  and  still  the  implication  is  strict  and  material,  but  not  formal. 
"All  men  are  mortal  and  Socrates  is  a  man"  does  not  formally  imply  "Soc- 
rates is  mortal".12  If  the  "proper"  meaning  of  "implies"  is  one  in  which 
"Socrates  is  a  man"  really  and  truly  implies  "Socrates  is  mortal",  or  one 
in  which  "All  men  are  mortal  and  Socrates  is  a  man"  really  and  truly 
implies  "Socrates  is  mortal",  then  formal  implication  is  not  that  proper 
meaning.  However  much  any  formal  implication  may  lie  behind  and 
support  such  an  inference,  it  cannot  state  it. 

One  further  consideration  is  worthy  of  note:  If  Tlx((px  c  \j/x)  be  restricted 
to  o;'s  which  exist,  then  it  will  denote  such  propositions  as  "'x  is  an  Ameri- 
can silver  coin  of  1915'  implies  'x  has  a  milled  edge'";  '"x  is  a  Monday 
of  last  March'  implies  'z  is  a  rainy  day ' ";  "x  has  horns  and  divided  hoofs' 
implies  'x  chews  a  cud' ".  In  other  words  it  will  denote  relations  which  are 
"contingent",  and  due  to  "coincidence".  It  may  be  doubted  whether 
such  relations  are  "properly"  implications.  But  upon  this  question  the 
reader  will  very  likely  find  himself  in  doubt.  What  we  regard  as  the 
reason  for  this  doubt  will  be  pointed  out  later. 

The  strict  implication,  p  -J  q,  means  "It  is  impossible  that  p  be  true 

11 1,  p.  21.     We  render  the  symbolism  of  this  passage  in  our  own  notation. 
12  It  may  be  objected  that  the  calculus  gives  the  formal  implication 

n^,,  ^  { [U,.(<fX  c  $x)  x  <pz]  c  tz} 

which  is  the  formal  implication  of  \{/z  by  Hx(<fixc  \f/x)  x  <pz,  which  is  required.  But  this  is 
not  what  is  required.  The  variables  for  all  values  of  which  this  proposition  is  asserted  are 
<f  and  \(/,  not  x  and  z.  The  reader  will  grasp  the  point  if  he  specify  <p  and  ^  here,  and  then 
allow  them  to  vary  in  his  illustration. 


The  System  of  Strict  Implication  333 

and  q  false",  or  "p  is  inconsistent  with  the  denial  of  q".  Similarly  <f>x  -J  \l/x 
means  "It  is  impossible  that  <px  be  true  and  \f/x  false",  or  "the  assertion 
of  <px  is  inconsistent  with  the  denial  of  \{/x".  Some  explanation  of  "im- 
possible" or  "inconsistent"  may  seem  called  for  here.  These  terms  can 
either  of  them  be  explained  by  the  other,  but  one  or  the  other  must  be  taken 
for  granted.  Yet  the  following  observations  may  be  of  assistance:  An 
assemblage  or  set  of  propositions  may  be  such  that  all  of  them  can  be  true 
at  once.  They  are  mutually  compatible,  compossible,  consistent.  There 
may  be  more  than  one  such  set.  Whoever  denies  this  on  metaphysical 
grounds  must  assume  the  burden  of  proof.  And  whether,  in  fact,  the 
possible  and  the  actual,  the  consistent  and  the  concurrently  true-in-fact 
are  identical,  at  least  one  must  admit  that  our  concept  of  the  possible 
differs  from  our  concept  of  the  actual:  that  we  mean  by  "consistent"  some- 
thing different  from  "concurrently  true-in-fact".  Any  set  of  mutually 
consistent  propositions  may  be  said  to  define  a  "possible  situation"  or 
"case"  or  "state  of  affairs".  And  a  proposition  may  be  "true"  of  more 
than  one  such  possible  situation — may  belong  to  more  than  one  such  set. 
Whoever  understands  "possible  situation"  thereby  understands  "con- 
sistent propositions",  and  vice  versa.  And  whoever  understands  "im- 
possible situation"  understands  also  "inconsistent  propositions".  In 
these  terms,  we  can  translate  p  -i  q  by  "Any  situation  in  which  p  should  be 
true  and  q  false  is  impossible". 

But  "situation"  as  here  used  should  not  be  confused  with  Boole's 
"times-  when  A  is  true".  A  proposition,  once  true,  is  always  true.  A 
proposition  may  be  true  of  some  possible  "situations"  and  false  of  others, 
but  it  must  be  in  point  of  fact  either  simply  true  or  simply  false.  This  is 
what  constitutes  the  distinction  between  a  proposition  and  a  prepositional 
function  such  as  "a;  is  a  man".  This  last  is,  in  point  of  fact,  neither  true 
nor  false.13 

Of  special  interest  are  the  cases  of  strict  implication  in  which  more 
than  two  propositions  are  involved.  We  have  already  seen  that  strict 
triadic  relations  take  the  form  of  strict  dyads,  one  member  of  which  is 
itself  a  non-strict  or  material  dyad.  WThere  we  might  expect  p  -J  (q  -J  r), 
wre  have  instead  p  -{  (q  c  r}  or  p  q  -i  r.  Instead  of  p  o  (q  o  r)  we  have 
p  o  (q  r)  or  (p  g)  o  r.  We  may  now  discover  the  reason  for  this — the  reason 

13  On  the  other  hand,  it  is  impossible  to  deny  that  a  proposition  may  be  true  of  some 
"situations",  and  false  of  others  unless  one  is  prepared  to  maintain  that  whatever  assertion 
can  be  referred  to  different  possible  "circumstances",  is  not  a  proposition.  And  whoever 
asserts  this  must,  to  be  consistent,  recognize  that  there  is  only  one  true  proposition,  the 
whole  of  the  truth,  the  assertion  of  all-fact,  the  Hegelian  Idee. 


334  A  Survey  of  Symbolic  Logic 

not  in  mathematical-wise  but  in  terms  of  common  sense  about  inference. 
Suppose  that  p,  q,  and  the  negation  of  r,  form  an  inconsistent  set.  They 
cannot  all  be  true  of  any  possible  situation.  We  have  symbolized  this  by 
-(poqo-r). 

(poq  o-r)  =  -[p  o  (q  -r)]  =  p  4  -(q  -r)  =  p  -t  (q  c  r)  =  p  q  -I  r 

If  p,  q,  and  -r  form  an  inconsistent  set  and,  in  point  of  fact  p  and  q  are  both 
true,  then  r  must  be  true  also.  So  much  is  quite  clear.  The  inference 
from  (p  q)  to  r  is  strict.  But  suppose  p,  q,  and  -r  cannot  all  be  true  in 
any  possible  situation  and  suppose  (in  the  actual  situation)  p  only  is  known 
to  be  true.  We  can  then  conclude  that  "If  q  is  true,  r  is  true". 

-(p  o  q  o  -r)  =  p  -J  (q  c  r) 

This  inference  is  also  strict,  but  our  symbolic  equivalents  tell  us  that  this 
"If  .  .  .  ,  then  .  .  . "  is  not  itself  a  strict  implication;  it  is  qcr,  a 
material  implication.  That  is  the  puzzle;  why  is  it  not  strict  like  the 
other?  The  answer  is  simple.  If  p,  q,  and  -r  cannot  all  be  true  in  any 
possible  situation  and  if  p  is  true  of  the  actual  situation,  it  follows  that 
q  and  -r  are  not  both  true  of  the  actual  situation,  that  is,  -(q  -r),  but  it 
does  not  follow  that  q  and  -r  cannot  both  be  true  in  some  other  possible 
situation  (in  which  p  should  be  false)  —it  does  not  follow  that  q  and  -r  are 
inconsistent,  that  -(qo-r).  Consequently  it  does  not  follow  that  q  •*  r, 
that  q  strictly  implies  r.  If,  then,  we  begin  with  an  a  priori  truth  (holding 
for  all  possible  situations),  that  p,  q,  and  -r  form  an  inconsistent  set,  and 
to  this  add  the  (empirical)  premise  "p  is  true",  we  get,  as  a  strict  con- 
sequence, the  proposition  "If  q  is  true,  r  is  true".  But  the  truth  of  this 
consequence  is  confined  to  the  actual  situation,  like  the  premise  p.  If,  in 
this  case,  we  go  on  and  infer  r  from  q,  our  inference  may  be  said  to  be  valid 
because  the  additional  premise,  p,  required  to  make  it  strict,  is  taken  for 
granted.  The  inference  depends  on  p  q  -J  r.  Or  we  may,  if  we  prefer, 
describe  it  as  an  inference  based  on  material  implication,  which  is  valid 
because  it  is  confined  to  the  actual  situation.  Much  of  our  reasoning  is 
of  this  type.  We  state,  or  have  explicitly  in  mind,  only  some  of  the  premises 
which  are  required  to  give  the  conclusion  strictly.  We  have  omitted  or 
forgotten  the  others,  because  they  are  true  and  are  taken  for  granted. 
In  this  sense,  much  of  our  reasoning  may  be  said  to  make  use  of  material 
or  formal  implications.  This  is  probably  the  source  of  our  doubts  whether 
such  propositions  as  '"x  is  an  American  silver  coin  dated  1915'  implies 
'x  has  a  milled  edge'",  and  "'x  has  horns  and  divided  hoofs'  implies  lx 


The  System  of  Strict  Implication  335 

chews  a  cud'"  represent  what  are  "properly"  called  implications.  In 
such  cases,  the  reasoning  is  valid  only  if  the  missing  premises,  which  would 
render  the  implication  strict,  are  capable  of  being  supplied. 

The  case  where  two  premises  are  strictly  required  for  inference  is  typi- 
cal of  all  those  which  require  more  than  one.  Where  three,  p,  q,  and  r, 
are  required  for  a  conclusion,  s,  we  have 

p  (q  r)  -J  s  =  p  -J  (q  r  c  s)  =  p  -{  [q  c  (r  c  s)] 

And  similar  equations  hold  where  four,  five,  etc.,  premises  are  required 
for  a  conclusion.  Only  the  main  implication  is  strict.  In  other  words,  a 
strict  implication  may  be  complex  but  is  always  dyadic. 

Another  significant  property  of  strict  implication,  as  opposed  to  material 
implication  or  formal  implication,  is  that  if  we  have  "(x  is  a  man'  strictly 
implies  'z  is  a  mortal'",  we  have  likewise  "'Socrates  is  a  man'  strictly 
implies  'Socrates  is  a  mortal'",  and  vice  versa.  Propositions  strictly  imply 
each  other  when  and  only  when  any  corresponding  prepositional  functions 
similarly  imply  one  another.  According  to  this  view  of  implication, 
"'Socrates  is  a  man'  implies  'Socrates  is  a  mortal'"  is  not  simply  felt  to 
be  the  kind  of  relation  upon  which  most  inference  depends:  it  is  the  rela- 
tion upon  which  all  inference  does  depend.  Strict  implication  is  the 
symbolic  representative  of  an  inference  which  holds  equally  well  whether 
its  terms  are  propositions  or  prepositional  functions. 

One  further  item  concerning  the  properties  of  strict  implication  has  to 
do  with  the  analogues  of  the  "peculiar"  propositions  of  Material  Impli- 
cation. These  analogues  are  themselves  somewhat  peculiar: 

3-52     ~p  -{  (p  -4  q)     If  p  is  impossible,  then  p  implies  any  proposition,  q. 
3-55     — p  -J  (q  -J  p)     If  p  is  necessarily  true,  then  p  is  implied  by  any 
proposition,  q. 

These  two  are  the  critical  members  in  this  class  of  propositions:  the  re- 
mainder follow  from  them  and  are  of  similar  import.  In  the  "proper" 
sense  of  "implies",  does  an  absurd,  not-self-consistent  proposition  imply 
anything  and  everything?  A  part  of  the  answer  is  contained  in  the 
observation  that  "necessary"  and  "impossible"  in  every-day  use  are 
commonly  hyperbolical  and  no  index.  No  proposition  is  "impossible" 
in  the  sense  of  ~p  except  such  as  imply  their  own  contradiction;  and  no 
proposition  is  "necessary"  in  the  sense  of  ~ -p  unless  its  negation  is  self- 
contradictory.  Again,  the  implications  of  an  absurd  proposition  are  no 
indication  of  what  would  be  true  if  that  absurd  proposition  were  true. 


33G  A  Survey  of  Symbolic  Logic 

It  is  the  nature  of  an  absurd  proposition  that  it  is  not  logically  conceivable 
that  it  should  be  true  under  any  possible  circumstances.  And,  finally, 
we  can  demonstrate  that,  in  the  ordinary  sense  of  "implies",  an  impossible 
proposition  implies  anything  and  everything.  It  will  be  granted  that  in 
the  "proper"  sense  of  "implies",  (1)  "p  and  q  are  both  true"  implies  " q  is 
true".  And  it  will  be  granted  that  (2)  if  two  premises  p  and  q  imply  a 
conclusion,  r,  and  that  conclusion,  r,  is  false,  while  one  of  the  premises, 
say  p,  is  true,  then  the  other  premise,  q,  must  be  false.  That  is,  if  "All 
men  are  liars"  and  "John  Blank  is  a  man"  together  imply  "John  Blank 
is  a  liar",  but  "John  Blank  is  a  liar"  is  false,  while  "John  Blank  is  a  man" 
is  true,  then  the  other  premise,  "All  men  are  liars",  must  be  false.  And  it 
will  be  granted  that  (3)  If  the  two  propositions,  p  and  q,  together  imply  r, 
and  r  implies  s,  then  p  and  q  together  imply  s.  These  three  principles  being 
granted,  it  follows  that  if  q  implies  r,  the  impossible  proposition  "q  is  true 
but  r  false"  implies  anything  and  everything.  For  by  (1)  and  (3),  if  q 
implies  r,  then  "p  and  q  are  both  true"  implies  r.  But  by  (2),  if  "p  and  q 
are  both  true"  implies  r,  "q  is  true  but  r  is  false"  implies  "p  is  false". 
Hence  if  q  implies  r,  then  "q  is  true  but  r  is  false"  implies  the  negation  of 
any  proposition,  p.  And  since  p  itself  may  be  negative,  this  impossible 
proposition  implies  anything.  "Today  is  Monday"  implies  "Tomorrow 
is  Tuesday".  Hence  "Today  is  Monday  and  the  moon  is  not  made  of 
green  cheese  "  implies  "Tomorrow  is  Tuesday".  Hence  "Today  is  Monday 
but  tomorrow  is  not  Tuesday"  implies  "It  is  false  that  the  moon  is  not 
made  of  green  cheese",  or  "The  moon  is  made  of  green  cheese". 

This  may  be  taken  as  an  example  of  the  fact  that  an  absurd  proposition 
implies  any  proposition.  It  should  be  noted  that  the  principles  of  the 
demonstration  are  quite  independent  of  anything  we  have  assumed  about 
strict  implication,  though  they  accord  with  our  assumptions. 

We  shall  now  demonstrate:  first,  that  there  are  a  considerable  class  of 
propositions  which  imply  their  own  contradiction  and  are  thus  impossible, 
and  a  class  of  propositions  which  are  implied  by  their  own  denial  and  are 
thus  necessary;  and  second,  that  an  impossible  proposition  implies  any 
proposition,  and  a  necessary  proposition  is  implied  by  any.  These  proofs 
wrill  be  similarly  free  from  any  necessary  appeal  to  symbolism,  making  use 
only  of  indubitable  principles  of  ordinary  logic. 

Any  proposition  which  should  witness  to  the  falsity  of  a  law  of  logic, 
or  of  any  branch  of  mathematics,  implies  its  own  contradiction  and  is 
absurd,  "p  implies  p"  is  a  law  of  logic;  and  may  be  used  as  an  example. 


The  System  of  Strict  Implication  337 

In  general,  any  implication,  "p  implies  <?,"  is  shown  false  by  the  fact  that 
"p  is  true  and  q  is  false".  Thus  the  law  "p  implies  p"  would  be  disproved 
by  the  discovery  of  any  proposition  p  such  that  "p  is  true  and  p  is  false". 
This  is,  then,  an  impossible  proposition,  "p  is  true  and  p  is  false"  implies 
its  own  negation,  which  is  "At  least  one  of  the  two,  not-p  and  p,  is  true". 
For  "p  is  true  and  p  is  false"  implies  "p  is  true".  And  "p  is  true"  implies 
"At  least  one  of  the  two,  p  and  q,  is  true".  And  not-p,  ('p  is  false,"  may 
be  this  q.  Hence  "p  is  true"  implies  "At  least  one  of  the  two,  p  and  not-p, 
is  true".  Hence  "p  is  true  and  p  is  false"  implies  "At  least  one  of  the  two, 
p  and  not-p,  is  true".  The  negation  of  "q  is  true  and  r  is  false"  is  "At 
least  one  of  the  two,  r  and  not-g,  is  true".  If  p  here  replace  both  q  and  r, 
we  have  as  the  negation  of  "p  is  true  and  p  is  false",  "At  least  one  of  the 
two,  p  and  not-p,  is  true".  And  it  is  this  which  "p  is  true  and  p  is  false" 
has  been  shown  to  imply. 

Merely  for  purposes  of  comparison,  we  resume  this  proof  in  the  symbols 
of  Strict  Implication: 

p  -J  p,  and  p  -J  p  =  ~(p  -p).     Hence  (p  -p)  is  an  impossible  proposition. 
By  the  principle  p  q  -J  p,  we  have  p-plp.  (1) 

And  by  the  principle  q  -{ p  +  q,  we  have  p  -J  (-p  +  p).  (2) 

By  the  principle  (p  •*  g)  (q  H  r)  -J  (p  -J  r),  this  gives  (1)  x  (2) 

p-p*  (-p  +  p) 

But  (-p  +  p)  =  -(p  -p).     Hence  p  -p  -i  -(p  -p). 

• 
This  is  only  one  illustration  of  a  process  which  might  be  carried  out  in 

any  number  of  cases.  Take  any  one  of  the  laws  of  Strict  Implication  and 
transform  it  into  a  form  which  has  the  prefix,  ~.  For  example, 

p  q  4  q  p  =  ~[(p  q)  -(q  p)] 

The  impossible  proposition  thus  discovered,  in  the  example  [(p  q)  -(q  p)], 
can  always  be  shown  to  imply  its  own  negation.  The  reader  will  easily  see 
how  this  may  be  done.  Such  illustrations  are  quite  generally  too  complex 
to  be  followed  through  without  the  aid  of  symbolic  abbreviation,  but  only 
the  principles  of  ordinary  logic  are  necessary  for  the  proofs. 

Wherever  we  find  an  impossible  proposition,  wre  find  a  necessary  propo- 
sition, its  negation.  For  example,  "At  least  one  of  the  two,  p  and  not-p, 
is  true"  is  a  necessary  proposition.  We  have  just  demonstrated  that  it  is 
implied  by  its  own  denial. 

(Some  logicians  have  been  inclined  of  late  to  deny  the  existence  of 

23 


338  A  Survey  of  Symbolic  Logic 

necessary  propositions  and  of  impossible,  or  self-contradictory,  propositions. 
We  beg  their  attention  to  the  above,  and  request  their  criticisms.) 

We  shall  now  prove  that  every  impossible  proposition — i.  e.,  every 
proposition  which  implies  its  own  negation — implies  anything  and  every- 
thing. If  p  implies  not-p,  then  p  implies  any  proposition,  q.  We  have 
already  shown  that  if  q  implies  r,  then  "q  is  true  and  r  is  false"  implies  any 
proposition.  Hence  if  p  implies  not-p,  "p  is  true  but  not-p  is  false", 
that  is,  "p  is  true  and  p  is  true",  implies  any  proposition,  q.  But  p  is 
equivalent  to  "p  is  true  and  p  is  true".  Hence  if  p  implies  not-p,  p  implies 
any  proposition,  q. 

Any  necessary  proposition,  i.  e.,  any  proposition,  q,  whose  denial,  not-g, 
implies  its  own  negation,  is  implied  by  any  proposition,  r.  This  follows 
from  the  above  by  the  principle  that  if  p  implies  q,  then  "  q  is  false  "  implies 
"p  is  false".  In  the  theorem  just  proved,  "If  p  implies  not-p,  then  p 
implies  any  proposition,  q",  let  p  be  "not-g",  and  q  be  "not-r".  We 
then  have  "If  not-g  implies  q,  then  not-g  implies  any  proposition,  not-r". 
And  if  not-g  implies  not-r,  then  "not-r  is  false"  implies  "not-q  is  false", 
i.  e.,  r  implies  q.  Hence  if  not-g  implies  q,  then  any  proposition,  r,  implies  q. 

But  "a  man  convinced  against  his  will  is  of  the  same  opinion  still". 
In  what  honest-to-goodness  sense  are  the  "necessary"  principles  of  logic 
and  mathematics  implied  by  any  proposition?  The  answer  is:  In  the 
sense  of  presuppositions.  And  what,  precisely,  is  that?  Any  principle,  A, 
may  be  said  to  be  presupposed  by  a  proposition,  B,  if  in  case  A  were  false, 
B  must  be  false.  If  a  necessary  principle  were  false,  anything  to  which 
it  is  at  all  relevant  would  be  false,  because  the  denial  of  such  a  principle, 
being  an  impossible  proposition,  implies  the  principle  itself.  And  where  a 
principle  and  its  negative  are  both  operative  in  a  system,  anything  which 
is  proved  is  liable  to  disproof.  Imagine  a  system  in  which  there  are  con- 
tradictory principles  of  proof.  That  the  chaotic  results  which  would  ensue 
are  not,  in  fact,  valid,  requires  as  presuppositions  the  truth  of  the  necessary 
laws  of  the  system.  These  laws — those  strictly  "necessary" — are  always 
logical  in  their  significance.  The  logic  of  "presupposition"  is,  in  fact,  a 
very  pretty  affair — we  have  no  more  than  suggested  its  character  here. 
The  time-honored  principles  of  rationalism  are  thoroughly  sound  and 
capable  of  the  most  rigid  demonstration,  however  much  the  historic  rational- 
ists have  stretched  them  to  cover  what  they  did  not  cover,  and  otherwise 
misused  them. 

In  this  respect,  then,  in  which  the  laws  of  Strict  Implication  seemed 


The  System  of  Strict  Implication  339 

possibly  not  in  accord  with  the  "  proper  "  sense  of  "  implies  ",  we  have  demon- 
strated that  they  are,  in  fact,  required  by  obviously  sound  logical  principles, 
though  in  ways  which  it  is  easy  to  overlook. 

It  may  be  urged  that  every  demonstration  we  have  given  shows  not 
only  that  impossible  propositions  imply  anything  and  necessary  proposi- 
tions are  implied  by  anything,  but  also  that  a  false  proposition  implies, 
anything,  and  a  true  proposition  is  implied  by  anything.  The  answer  is- 
that  an  impossible  proposition  is  false,  of  course,  and  a  necessary  proposition 
is  true.  But  if  anyone  think  that  this  validates  the  doubtful  theorems  of 
Material  Implication,  it  is  incumbent  upon  him  to  show  that  some  proposi- 
tion that  is  false  but  not  impossible  implies  anything  and  everything,  and 
that  some  proposition  which  is  true  but  not  necessary  is  implied  by  all  propo- 
sitions. And  this  cannot  be  done. 

We  shall  not  further  prolong  a  tedious  discussion  by  any  special  plea 
for  the  " propriety"  of  strict  implication  as  against  material  implication 
and  formal  implication.  Anyone  who  has  read  through  so  much  technical 
and  uninteresting  matter  has  demonstrated  his  right  and  his  ability  to 
draw  his*  own  conclusions. 


CHAPTER    VI 

SYMBOLIC  LOGIC,  LOGISTIC,  AND  MATHEMATICAL 

METHOD 

I.    GENERAL  CHARACTER  OF  THE  LOGISTIC  METHOD.    THE  "ORTHODOX" 

VIEW 

The  method  of  any  science  depends  primarily  upon  two  factors,  the 
medium  in  which  it  is  expressed  and  the  type  of  operations  by  which  it 
is  developed.  "Logistic"  may  be  taken  to  denote  any  development  of 
scientific  matter  which  is  expressed  exclusively  in  ideographic  language  and 
uses  predominantly  (in  the  ideal  case,  exclusively)  the  operations  of  sym- 
bolic logic.  Though  this  definition  would  not  explicitly  include  certain 
cases  of  what  would  undoubtedly  be  called  "logistic",  and  we  shall  wish 
later  to  present  an  alternative  view,  it  seems  best  to  take  this  as  our  point 
of  departure. 

"Modern  geometry"  differs  from  Euclid  most  fundamentally  by  the 
fact  that  in  modern  geometry  no  step  of  proof  requires  any  principle  except 
the  principles  of  logic.1  It  was  the  extra-logical  principles  of  proof  in 
Euclidean  geometry  and  other  branches  of  mathematics  which  Kant 
noted  and  attributed  to  the  "pure  intuition"  of  space  (and  time)  as  the 
source  of  "synthetic  judgments  a  priori"  in  science.  The  character  of 
space  (or  of  time),  as  apprehended  a  priori,  carries  the  proof  over  places 
where  the  more  general  principles  of  logic — "analysis" — cannot  take  it. 
Certain  operations  of  thought  are,  thus,  accepted  as  valid  in  geometry 
because  geometry  is  thought  about  space,  and  these  transformations  are 
valid  for  spatial  entities,  though  they  might  not  be  valid  for  other  things. 
The  principal  impetus  to  the  modern  method  in  geometry  came  from  the 
discovery  of  non-Euclidean  systems  which  must  necessarily  proceed,  to 
some  extent,  without  the  aid  of  such  space  intuitions,  a  priori  or  otherwise. 
And  the  perfection  of  the  modern  method  is  attained  when  geometry  is 
entirely  freed  from  dependence  upon  figures  or  constructions  or  any  appeal 

1  In  the  opinion  of  most  students,  Euclid  himself  sought  to  give  his  proofs  the  rigorous 
character  which  those  of  modern  geometry  have,  and  the  difference  of  the  two  systems  is 
in  degree  of  attainment  of  this  ideal.  But  Euclid's  successors  introduced  methods  which 
still  further  depended  upon  intuition. 

340 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  341 

to  the  perceptual  character  of  space.  When  geometry  is  thus  freed  from 
this  appeal  to  intuition  or  perception,  the  methods  of  proof  are  simply 
those  which  are  independent  of  the  nature  of  the  subject  matter  of  the  science — 
that  is,  the  methods  of  logic,  which  are  valid  for  any  subject  matter.2 

Coincidently  with  this  alteration  of  method  comes  another  change — 
geometry  is  now  abstract.  If  nothing  in  the  proofs  depends  upon  the  fact 
that  the  terms  denote  certain  spatial  entities,  then,  whatever  may  be  meant 
by  "point",  "plane",  "triangle",  "parallel",  etc.,  if  the  assumptions  be 
true,  then  the  theorems  will  be.  Or  in  any  sense  in  which  the  assumptions 
can  be  asserted,  in  that  same  sense  all  the  consequences  of  them  can  be 
asserted.  The  student  may  carry  in  his  mind  any  image  of  "triangle" 
or  "parallels"  which  is  consistent  with  the  propositions  about  them.  More 
than  this,  even  the  geometrical  relations  asserted  to  hold  between  "points", 
"lines",  etc.,  may  be  given  any  denotation  which  is  consistent  with  the 
properties  assigned  to  them.  In  general,  this  means  for  relations,  that  any* 
meaning  may  be  assigned  which  is  consistent  with  the  type  of  the  relation — 
e.  g.,  transitive  or  intransitive,  symmetrical  or  unsymmetrical,  one-one  or 
one-many,  etc. — and  with  the  distributions  of  such  relations  in  the  system. 

Essentially  the  same  evolution  has  taken  place  in  arithmetic,  or  "alge- 
bra ".  Any  reference  to  the  empirical  character  of  tally  marks  or  collections 
of  pebbles  has  become  unnecessary  and  naive.  The  " indefinables "  of 
arithmetic  are  specified,  very  likely,  as  "A  class,  K,  of  elements,  a,  b,  c, 
etc.,  and  a  relation  (or  'operation')  +".  Definitions  have  come  to  have 
the  character  of  what  Kant  called  "transcendental  definitions" — that  is 
to  say,  they  comprehend  those  properties  which  differentiate  the  entity,  to 
be  defined  by  its  logical  relations,  not  those  which  distinguish  it  for  sense 
perception.  The  real  numbers,  for  instance,  no  longer  denote  the  possible 
lengths  of  a  line,  but  are  the  class  of  all  the  "cuts"  that  can  be  made  (logi- 

2 1  cannot  pass  over  this  topic  without  a  word  of  protest  against  the  widespread  notion 
that  the  development  of  modern  geometry  demonstrates  the  falsity  of  Kant's  Transcendental 
Aesthetik.  It  does  indeed  demonstrate  the  falsity  of  Kant's  notion  that  such  "synthetic" 
principles  are  indispensable  to  mathematics.  But,  in  general,  it  is  accurate  to  say  that 
Kant's  account  is  concerned  with  the  source  of  our  certainty  about  the  world  of  nature, 
not  with  the  methods  of  abstract  science  which  did  not  exist  in  his  day.  Nothing  is  more 
obvious  than  that  the  abstractness  of  modern  geometry  comes  about  through  definitely 
renouncing  the  thing  which  Kant  valued  in  geometry — the  certainty  of  its  applicability 
to  our  space.  When  geometry  becomes  abstract,  the  content  of  the  science  of  space  splits 
into  two  distinct  subjects:  (1)  geometry,  and  (2)  the  metaphysics  of  space,  which  is  con- 
cerned with  the  application  of  geometry.  This  second  subject  has  been  much  discussed 
since  the  development  of  modern  geometry,  usually  in  the  skeptical  or  "pragmatic"  spirit 
(vide  Poincare1).  But  it  is  possible — and  to  me  it  seems  a  fact — that  Kant's  basic  argu- 
ments are,  with  qualifications,  capable  of  being  rehabilitated  as  arguments  concerning  the 
certainty  of  our  knowledge  of  the  phenomenal  world,  i.  e.  as  a  metaphysics  of  space. 


342  A  Survey  of  Symbolic  Logic 

cally  specified)  in  a  dense,  denumerable  series,  of  the  type  of  the  series  of 
rationals. 

Thus  abstractness  and  the  rigorously  deductive  method  of  development 
have  more  and  more  prevailed  in  the  most  careful  presentations  of  mathe- 
matics. When  these  are  completely  achieved,  a  mathematical  system  becomes 
nothing  more  nor  less  than  a  complex  logical  structure.3  Consider  any  two 
mathematical  systems  which  have  been  given  this  ideal  mathematical  form. 
They  will  not  be  distinguished  by  the  entities  which  form  their  "subject 
matter",  for  the  terms  of  neither  system  have  any  fixed  denotation.  And 
they  will  not  be  distinguished  by  the  operations  by  means  of  which  they 
are  developed,  for  the  operations  will,  in  both  cases,  be  simply  those  of 
logical  demonstration. 

A  word  of  caution  upon  the  meaning  of  "operation"  is  here  necessary. 
It  is  exactly  by  the  elimination  of  all  peculiarly  mathematical  operations 
that  a  system  comes  to  have  the  rigorously  deductive  form.  For  the 
grocer  who  represents  his  putting  of  one  sack  of  sugar  with  another  sack 
by  25  +  25  =  50,  [+]  is  a  symbol  of  operation.  For  the  child  who  learns 
the  multiplication  table  as  a  means  to  the  manipulation  of  figures,  [X] 
represents  an  operation,  but  in  any  rigorously  deductive  development  of 
arithmetic,  in  Dede kind's  Was  sind  und  ivas  sollen  die  Zahlen,  or  Hunting- 
ton's  "Fundamental  Laws  of  Addition  and  Multiplication  in  Elementary 
Algebra",  [+]  and  [X]  are  simply  relations.  An  operation  is  something 
done,  performed.  The  only  things  performed  in  an  abstract  deductive  system 
are  the  logical  operations — variables  are  not  added  or  multiplied.  But, 
unfortunately,  such  relations  as  [+]  and  [X]  are  likely  to  be  still  spoken 
of  as  "operations".  Hence  the  caution. 

Since  abstract  mathematical  systems  do  not  differ  by  any  fixed  meaning 
of  their  terms,  and  since  they  are  not  distinguished  through  their  operations, 
they  will  be  different  from  one  another  only  with  respect  to  the  relations 
of  their  terms,  and  probably  also  in  certain  relations  of  a  higher  order — 
relations  of  relations.  And  the  relations,  being  likewise  abstract,  wrill 
differ,  from  system  to  system,  only  in  type  and  in  distribution  in  the  systems; 
that  is,  any  two  systems  will  differ  only  as  types  of  logical  order. 

3  M.  Fieri,  writing  of  "La  Ge'omStrie  envisaged  comme  systems  purement  logique", 
says:  "Je  tiens  pour  assur6  que  cette  science,  dans  ces  parties  les  plus  elevens  comme 
dans  les  plus  modestes,  va  en  s'affirment  et  en  consolidant  de  plus  en  plus  comme  I'etude 
d'un  certain  ordre  de  relations  logiques;  en  s'affranchissant  peu  a  peu  des  liens  qui  1'attachent 
a  1'intuition,  et  en  revetant  par  suite  la  forme  et  les  qualit^s  d'un  science  ideale  purement 
deductive  et  abstraite,  comme  VArithmetique".  (Bibliotheque  du  congres  Internationale  de 
Philosophic,  in,  368.) 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  343 

The  connection  between  abstract  or  "pure"  mathematics  and  logistic 
is,  thus,  a  close  one.  But  the  two  cannot  be  simply  identified.  For  the 
logical  operations  by  which  the  mathematical  system  is  generated  from  its 
assumptions  may  not  themselves  be  expressed  in  ideographic  symbols. 
Ordinarily  they  are  not:  there  are  symbols  for  "four"  and  "congruent", 
"triangle"  and  "plus",  but  the  operations  of  proof  are  expressed  by  "If 
.  .  .  then  .  .  .",  "Either  .  .  .  or  .  .  .,"  etc.  Only  when  the  logical 
operations  also  are  expressed  in  ideographic  symbols  do  we  have  logistic. 
In  other  words,  all  rigorously  deductive  mathematics  gets  its  principles  of 
operation  from  logic;  logistic  gets  its  principles  of  operation  from  symbolic 
logic.  Thus  logistic,  or  the  logistic  development  of  mathematics,  is  a  name 
for  abstract  mathematics  the  logical  operations  of  whose  development 
are  represented  in  the  ideographic  symbols  of  symbolic  logic. 

Certain  extensions  of  symbolic  logic,  as  we  have  reviewed  it,  are  needed 
for  the  satisfactory  expression  of  these  mathematical  operations— particu- 
larly certain  further  developments  of  the  logic  of  relations,  and  the  theory 
of  what  are  called  "descriptions"  in  Principia  Mathematica.  But  these 
necessary  additions  in  no  wise  affect  what  has  been  said  of  the  relation 
between  symbolic  logic  and  the  logistic  development  of  mathematics. 

II.     Two   VARIETIES   or  LOGISTIC   METHOD:    PEANO'S  Formulaire  AND 
Principia  Mathematica.     THE  NATURE  OF  LOGISTIC  PROOF 

The  logistic  method  is,  then,  a  universal  method,  applicable  to  any 
sufficiently  coordinated  body  of  exact  knowledge.  And  it  gives,  in  mathe- 
matics, a  most  precise  and  compact  development,  displaying  clearly  the 
type  of  logical  order  which  characterizes  the  system.  However,  there  are 
certain  variations  of  the  logistic  method,  and  systems  so  developed  may 
differ  widely  from  one  another  in  ways  which  have  nothing  directly  to  do 
with  the  type  and  distribution  of  relations.  One  most  important  difference 
has  to  do  with  the  degree  to  which  the  analysis  of  terms  is  carried  out. 
"Number,"  for  example,  may  be  taken  simply  as  a  primitive  idea,  or  it 
may  be  defined  in  terms  of  more  fundamental  notions.  And  these  notions 
may,  in  turn,  be  defined.  The  length  to  which  such  analysis  is  carried,  is 
an  important  item  in  determining  the  character  of  the  system.  Correl- 
atively,  relations  such  as  [+]  and  [X]  may  be  taken  as  primitive,  or  they 
may  be  defined.  And,  finally,  the  fundamental  propositions  which  generate 
the  system  may  be  simply  assumed  as  postulates,  or  they  may,  by  the  analy- 
sis just  mentioned,  be  derived  from  those  of  a  more  elementary  discipline. 


344  A  Survey  of  Symbolic  Logic 

In  general,  the  analysis  of  "terms"  and  of  relations  and  the  derivation  of 
fundamental  propositions  go  together.  An,d  the  use  of  this  analytic 
method  requires,  to  some  extent  at  least,  a  hierarchy  of  subjects,  with 
symbolic  logic  as  the  foundation  of  the  whole. 

To  illustrate  these  possible  differences  between  logistic  systems,  it  will 
be  well  to  compare  two  notable  developments  of  mathematics:  Formulaire 
de  Mathematiques*  of  Peano  and  his  collaborators,  and  Principia  Mathe- 
matica  of  Whitehead  and  Russell.  These  two  are  by  no  means  opposites 
in  the  respects  just  mentioned.  Principia  Mathematica  represents  the 
farthest  reach  of  the  analytic  method,  having  no  postulates  and  no  primitive 
ideas  save  those  of  the  logic,  while  the  Formulaire  exhibits  a  partially  hier- 
archic, partially  independent,  relation  of  various  mathematical  branches.5 

For  example,  in  the  Formulaire,  the  following  primitive  ideas  are  assumed 
for  arithmetic,  which  immediately  succeeds  "mathematical  logic". 

NO  signifies  'number',  and  is  the  common  name  of  0,  1,  2,  etc. 

0  signifies  'zero'. 

+  signifies  'plus'.  If  a  is  a  number,  a  +  indicates  'the  number  suc- 
ceeding a'.6 

The  primitive  propositions,  or  postulates,  are  as  follows: 7 
1-0  NoeCls 
1-1  OeN<> 
1-2  aeNo.3.  a  +  eN0 

1  •  3  s  e  Cls  •Oes:aes.3f,.a  +  es:D.N0es 
1-4  a,  &  e  NO  .  a  +  =b  +  mO.a  =  b 
l-5a«No«3«a  +  -=0 

The  symbol  D  here  represents  ambiguously  "implies"  or  "is  contained 
in" — the  relation  c  of  the  Boole-Schroder  Algebra.  This  and  the  idea  of 
a  class,  "Cls",  and  the  e-relation,  are  defined  and  their  properties  demon- 
strated in  the  "mathematical  logic".  In  terms  of  these,  the  above  propo- 
sitions may  be  read: 

4  All  our  references  will  be  to  the  fifth  edition,  which  is  written  in  the  proposed  inter- 
national language,  Interlingua,  and  entitled  Formulario  Mathemalico,  Editio  v  (Tomo  v  de 
Formulario  complete). 

5  The  independence  of  various  branches  in  the  Formulaire  is  somewhat  greater  than  a 
superficial  examination  reveals.     Not  only  are  there  primitive  propositions  for  arithmetic 
and  geometry,  but  many  propositions  are  assumed  as  "definitions"  which  define  in  that 
discursive  fashion  in  which  postulates  define,  and  which  might  as  well  be  called  postulates. 
Observe,  for  example,  the  definitions  of  +  and  X,  to  be  quoted  shortly. 

6  Section  n,  §  1,  p.  27. 

7  Ibid. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  345 

1-0  NO  is  a  class,  or  'number'  is  a  common  name. 

I  •  1  0  is  a  number. 

1-2  If  a  is  a  number,  then  the  successor  of  a  is  a  number. 

1-3  If  s  is  a  class,  and  if  0  is  contained  in  s,  and  if,  for  every  a,  'a  is 
contained  in  s'  implies  'the  successor  of  a  is  contained  in  s',  then  N0  is 
contained  in  s  (every  number  is  a  member  of  the  class  s}. 

(1-3  is  the  principle  of  "mathematical  induction".) 

1-4  If  a  and  b  are  numbers,  and  if  the  successor  of  a  =  the  successor 
of  b,  then  a  =  b. 

1-5  If  a  is  a  number,  then  the  successor  of  a  =}=  0. 

The  numbers  are  then  defined  in  the  obvious  way:  1=0+, 2  =  1+, 
3  =  2+,  etc.8  The  relation  +,  which  differs  from  the  primitive  idea,  a  +, 
is  then  defined  by  the  assumptions: 9 

3-laeN0.3.a  +  0  =  a 
(If  a  is  a  number,  then  a  +  0  =  a.) 

3-2  a,  6  e  N0. 3.  a  +  (b  +)  =  (a  +  6)  + 

(If  a  and  b  are  numbers,  then  a  +  'the  successor  of  b'  =  'the  successor  of 
a  +  6'.) 

The  relation  X  is  defined  by: 10 

1-0  a,  b,  c  e  No.  a.  a  X  0  =  0 

1-01  a,  b,  c  €  No .  3 .  a  X  (b  +  1)  =  (a  X  6)  +  a 

It  will  be  clear  that,  except  for  the  expression  of  logical  relations,  such 
as  f  and  o ,  in  ideographic  symbols,  these  postulates  and  definitions  are  of 
the  same  general  type  as  any  set  of  postulates  for  abstract  arithmetic. 
A  class,  NO,  of  members  a,  b,  c,  etc.,  is  assumed,  and  the  idea  of  a  +,  "suc- 
cessor of  a".  The  substantive  notions,  "number"  and  "zero",  the  de- 
scriptive function,  "successor  of,"  the  relations  +  and  X,  are  not  analysed 
but  are  taken  as  simple  notions.11  However,  the  properties  which  numbers 
have  by  ^virtue  of  being  members  of  a  class,  N0,  are  not  taken  for  granted,  as 
would  necessarily  be  done  in  a  non-logistic  treatise — they  are  specifically 
set  forth  in  propositions  of  the  "mathematical  logic"  which  precedes. 
And  the  other  principles  by  which  proof  is  accomplished  are  similarly 
demonstrated.  Of  the  specific  differences  of  method  to  which  this  explicit- 
ness  of  the  logic  leads,  we  shall  speak  shortly. 

8  See  ibid.,  p.  29. 

9  Ibid. 

10  See  ibid.,  §  2,  p.  32. 

II  Peano  does  not  suppose  them  to  be  unanalyzable.    He  says  (p.  27):   "Quaesitione 
si  nos  pote  defini  No,  significa  si  nos  pote  scribe  aequalitate  de  forma,  No  =  expressione 
composito  per  signos  noto  ~  ~  7  ...  -,  quod  non  est  facile".     (This  was  written  after  the 
publication  of  Russell's  Principles  of  Mathematics,  but  before  Principia  Mathematical 


346  A  Survey  of  Symbolic  Logic 

In  Principia  Mathematica,  there  are  no  separate  assumptions  of  arith- 
metic, except  definitions  which  express  equivalences  of  notation  and  make 
possible  the  substitution  of  a  single  symbol  for  a  complex  of  symbols. 
There  are  no  postulates,  except  those  of  the  logic,  in  the  whole  work.  In 
other  words,  all  the  properties  of  numbers,  of  sums,  products,  powers,  etc., 
are  here  proved  to  be  what  they  are,  solely  on  account  of  what  number  is, 
what  the  relations  +  and  X  are,  etc.  Postulates  of  arithmetic  can  be' 
dispensed  with  because  the  ideas  of  arithmetic  are  thoroughly  analysed. 
The  lengths  to  which  such  analysis  must  go  in  order  to  derive  all  the  proper- 
ties of  number  solely  from  definitions  is  naturally  considerable.  We  should 
be  quite  unable,  within  reasonable  space,  to  give  a  satisfactory  account  of 
the  entities  of  arithmetic  in  this  manner.  In  fact,  the  latter  half  of  Volume 
I  and  the  first  half  of  Volume  II  of  Principia  Mathematica  may  be  said  to 
do  nothing  but  just  this.  However,  we  may,  as  an  illustration,  follow  out 
the  analysis  of  the  idea  of  "cardinal  number".  This  will  be  tedious  but, 
with  patience,  it  is  highly  instructive. 

We  shall  first  collect  the  definitions  which  are  involved,  beginning  with 
the  definition  of  cardinal  number  and  proceeding  backward  to  the  definition 
of  the  entities  in  terms  of  which  cardinal  number  is  defined,  and  then  to 
the  entities  in  terms  of  which  these  are  defined,  and  so  on.12 

*100-02    NC  =  D'Nc.        Df 

"Cardinal  number"  is  the  defined  equivalent  of  "the  domain  of  (the  rela- 
tion) Nc". 

*33-01     D  =  aR[a  =  x{(3.y}  .xRy].        Df 

"  D  "  is  the  relation  of  (a  class)  a  to  (a  relation)  R,  when  a  and  R  are  such 
that  a  is  (the  class)  x  which  has  the  relation  R  to  (something  or  other)  y. 
That  is,  "D"  is  the  relation  of  a  class  of  x's,  each  of  which  has  the  relation 
R  to  something  or  other,  to  that  relation  R  itself. 

*30-01     R'y  =  (TX)(xRy}.        Df 

" R'y"  means  "the  x  which  has  the  relation  R  to  y". 

Putting  together  this  definition  of  the  use  of  the  symbol  '  and  the 
definition  of  "D",  we  see  that  "D'Nc"  is  "the  x  which  has  the  relation 
D  to  Nc",  and  this  £  is  a  class  a  such  that  every  member  of  a  has  the  rela- 

12  The  place  of  any  definition  quoted,  in  Principia,  is  indicated  by  the  reference  number. 
The  "  translations"  of  these  definitions  are  necessarily  ambiguous  and  sometimes  inaccurate, 
and,  of  course,  any  "translation"  must  anticipate  what  here  follows — but  in  Principia 
precedes. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  347 

tion  Nc  to  something  or  other.     "D'R"  is  "the  domain  of  the  relation  R". 
If  "R"  be  "precedes",  then  "D'R"  will  be  "the  class  of  all  those  things 
which   precede   anything".     "Cardinal   number",    "NC,"   is   defined    as 
"D'Nc",  "the  domain  of  the  relation  Nc". 
We  now  turn  to  the  meaning  of  "Nc". 

*100-01     Nc  =  sm.        Df 

"  Nc  "  is  the  relation  of  the  class  of  referents  of  "  sm  "  to  "  sm  "  itself.  First, 
let  us  see  the  meaning  of  the  arrow  over  "sm". 

*32-01     R  =  ay{a  =  x(xRy)}.        Df 

" R"  is  "the  relation  of  a  to  y,  where  a  and  y  are  such  that  a  is  the  class 
of  x's,  each  of  which  has  the  relation  R  to  y".  If  " R"  be  "precedes", 
" R"  will  be  the  relation  of  the  class  "predecessors  of  y"  to  y  itself. 

Now  for  "sm".     We  shall  best  not  study  its  definition  but  a  somewhat 
simpler  proposition. 

*73-l      asm/3.  =  .  (3#)  .flel  -»1  -a  =  D'R.  (3  =  d'R 

"  a  sm  j8"  is  equivalent  to  "For  some  relation  R,  R  is  a  one-to-one  relation, 
while  a  is  the  domain  of  R  and  /3  is  the  converse-domain  of  R". 
We  have  here  anticipated  the  meaning  of  "d'R"  and  of  "1  — >  1". 

*33-02    a  =  $R[p  =  ${(3.x).xRy}].        Df 

"G"  is  "the  relation  of  (a  class)  /3  to  (a  relation)  R,  when  /3  and  R  are 
such  that  /3  is  the  class  of  y's,  for  each  of  which  (something  or  other)  x  has 
the  relation  R  to  y".  Comparing  this  with  the  definition  of  "D"  and  of 
"D'R"  above,  we  see  that  "Q'R",  the  converse-domain  of  R,  is  the  class 
of  those  things  to  which  something  or  other  has  the  relation  R.  If  " R" 
be  "precedes",  "Q.'R"  will  be  the  class  of  those  things  which  are  preceded 
by  something  or  other. 

*71-03     1  -»1  =  R(R"a'Rcl.R"D'Rcl).        Df 
This  involves  the  meaning  of  "R",  of  ",  and  of  "1 ". 
*32-02    #  =  $x{(3  =  y(xRy}}.        Df 

" R"  signifies  "the  relation  of  /3  to  x,  when  ft  and  x  are  such  that  j8  is  the 
class  of  y's  to  which  x  has  the  relation  R. 

*37-01     R"p  =  £{(30)  .yt-p.xRy}.        Df 

" R"P"  is  "the  class  of  x's  such  that,  for  some  y,  y  is  a  member  of  /3,  and  x 
has  the  relation  R  to  y.  In  other  words,  " R"P"  (the  R's  of  the  /3's)  is  the 


348  A  Survey  of  Symbolic  Logic 

class  of  things  which  have  the  relation  R  to  some  member  or  other  of  the 
class  j8.  If  "R"  be  "precedes",  "jR"/3"  will  be  the  class  of  predecessors  of 
all  (any)  members  of  0. 

With  the  help  of  this  last  and  of  preceding  definitions,  we  can  now  read 
*71-03.  "1  -»  1"  is  "the  class  (of  relations)  R,  such  that  whatever  has 
the  relation  R  to  any  member  of  the  class  of  things-to-which-anything-has- 
the-relation-^,  is  contained  in  1;  and  whatever  is  such  that  any  member 
of  the  class  of  those-things-which-have-the-relation-.R-to-anything  has  the 
relation  R  to  it,  is  contained  in  1 . "  Or  more  freely  and  intelligibly :  "  1  — »  1 " 
is  the  class  of  relations,  R,  such  that  if  a  R  ft  is  true,  then  a  is  a  class  of 
one  member  and  /3  is  a  class  of  one  member:  "1  — >  1"  is  the  class  of  all 
one-to-one  correspondences.  Hence  "  a  sm  /3 "  means  "  There  is  a  one-to- 
one  correspondence  of  the  members  of  a  with  the  members  of  /?.  "sm" 
is  the  relation  of  classes  which  are  (cardinally)  similar. 

The  analysis  of  the  idea  of  cardinal  number  has  now  been  carried  out 
until  the  undefined  symbols,  except  "1",  are  all  of  them  logical  symbols; — 
of  relations,  R;  of  classes,  a,  (3,  etc.;  of  individuals,  x,  y,  etc.;  of  prepo- 
sitional functions  such  as  xRy  [which  is  a  special  case  of  <p(x,  y)};  of 
"  <f>x  for  some  x",  (3  a:)  .  <px;  the  relations  c,  c,  and  =;  and  the  idea 
(i  x)(<px),  "the  x  for  which  px  is  true".  This  last  notion  occurs  in  various 
special  cases,  such  as  D'R,  R"p,  etc. 

"  1 "  is  also  defined  in  terms  which  reduce  to  these,  but  the  definitions 
involved  are  incapable  of  precise  translation — more  accurately,  ordinary 
language  is  incapable  of  translating  them. 

*52-01     1  =  a {(3s)  .  a  =  i'x\.        Df 

*51-01     i  =  L  .      Df 

*50-01     I=xy(x  =  y}.        Df13 

"/"  is  the  relation  of  identity;  "i"  is  the  class  of  those  things  which  have 
the  relation  of  identity  to  something  or  other;  and  "  1 "  is  the  class  of  such 
classes,  i.  e.,  the  class  of  all  classes  having  only  a  single  member.  Thus  the 
definition  of  "1"  is  given  in  terms  of  the  idea  of  individuals,  x  and  y,  of 
the  relation  = ,  of  classes,  and  the  idea  involved  in  the  use  of  the  arrow 
over  I,  which  has  already  been  analyzed.  This  definition  of  1  is  in  no 
wise  circular,  however  much  its  translation  may  suggest  that  it  is;  nor  is 
there  any  circularity  involved  in  the  fact  that  the  definition  of  cardinal 
number  requires  the  previous  definition  of  "  1 ". 

13  Strictly,  analysis  of  =,  which  differs  from  the  defining  relation,  [. . .  =  ...  Df],  is 
required.  But  the  lack  of  this  does  not  obscure  the  analysis,  so  we  omit  it  here. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  349 

We  have  now  completely  accomplished  the  analysis  of  the  idea  of  cardinal 
number  into  constituents  all  of  which  belong  to  mathematical  logic.  The 
important  significance  of  this  analysis  for  the  method  involved  we  must 
postpone  for  a  moment  to  discuss  the  definition  itself. 

If  we  go  back  over  these  definitions,  we  find  that  the  notion  of  cardinal 
number  can  now  be  defined  as  follows:  "Cardinal  number"  is  the  class  of 
all  those  classes  the  members  of  which  have  a  one-to-one  correspondence 
(with  members  of  some  other  class).  "Cardinal  number"  is  the  class  of 
all  the  cardinal  numbers;  and  a  cardinal  number  is  the  class  of  all  those 
classes  whose  members  have  a  one-to-one  correlation  with  the  members  of  a 
given  class.  This  is  definition  "in  extension".  We  most  frequently  think 
of  the  cardinal  number  of  a  class,  a,  as  a  property  of  the  class.  Definition 
in  extension  determines  any  such  property  by  logically  exhibiting  the  class 
of  all  those  things  which  have  that  property.  Thus  if  a  be  the  class  com- 
posed of  Henry,  Mary  and  John,  the  cardinal  number  of  a  will  be  deter- 
mined by  logically  exhibiting  all  those  classes  which  have  a  one-to-one 
correlation  with  the  members  of  a — i.  e.,  all  the  classes  with  three  members. 
"3"  will,  then,  be  the  class  of  all  classes  having  three  members;  "4",  the 
class  of  all  classes  of  four,  etc.  And  "cardinal  number"  in  general  will  be 
the  class  of  all  such  classes  of  classes. 

It  may  be  well  to  observe  here  also  that,  by  means  of  ideographic  sym- 
bols, we  can  represent  exactly,  and  in  brief  space,  ideas  which  could  not 
possibly  be  grasped  or  expressed  or  carried  in  mind  in  any  other  terms. 
Perhaps  the  reader  has  not  grasped  those  presented:  we  can  assure  him  it 
is  not  difficult  once  the  symbolism  is  clear.  And  if  the  symbolism  appals 
by  its  unfamiliarity,  we  would  call  attention  to  the  fact  that  the  number  of 
different  symbols  is  not  greater,  nor  is  their  meaning  more  obscure  than 
those  of  the  ordinary  algebraic  signs.  It  is  the  persistent  accuracy  of  the 
analysis  that  has  troubled  him;  far  be  it  from  us  to  suggest  that  we  do  not 
like  to  think  accurately. 

So  much  analysis  may  appeal  to  us  as  unnecessary  and  burdensome. 
But  observe  the  consequences  of  it  for  the  method.  When  "cardinal 
number"  is  defined  as  "D'Nc,"  all  the  properties  of  cardinal  number  follow 
from  the  properties  of  "D"  and  "Nc"  and  the  relation  between  these 
represented  by  '.  And  when  these  in  turn  are  defined  in  terms  of  "sm" 
and  the  idea  expressed  by  the  arrow,  and  so  on,  their  properties  follow  from 
the  properties  of  the  entities  which  define  them.  And  finally,  when  all  the 
constituents  of  "cardinal  number",  and  the  other  ideas  of  arithmetic 


350  A  Survey  of  Symbolic  Logic 

have  been  analyzed  into  ideas  which  belong  to  symbolic  logic,  all  the  propo- 
sitions about  cardinal  number  follow  from  these  definitions.  When  analysis 
of  the  ideas  of  arithmetic  is  complete,  all  the  propositions  of  arithmetic 
follow  from  the  definitions  of  arithmetic  together  with  the  propositions  of 
logic.  Now  in  Principia  Mathematica  it  is  found  possible  to  so  analyze  all 
the  ideas  of  mathematics.  Hence  the  whole  of  mathematics  is  proved 
from  its  definitions  together  with  the  propositions  of  logic.  And,  except 
the  logic,  wo  branch  of  mathematics  needs  any  primitive  ideas  or  postulates  of 
its  own.  It  is  thus  demonstrated  by  this  analysis  that  the  only  postulates 
and  primitive  ideas  necessary  for  the  whole  of  mathematics  are  the  postu- 
lates and  primitive  ideas  of  logic. 

In  the  light  of  this,  we  can  understand  Mr.  Russell's  definition  of 
mathematics : 14 

"Pure  Mathematics  is  the  class  of  all  propositions  of  the  form  'p  im- 
plies q'  where  p  and  q  are  propositions  containing  one  or  more  variables, 
the  same  in  the  two  propositions,  and  neither  p  nor  q  contains  any  constants 
except  logical  constants.  And  logical  constants  are  all  notions  definable 
in  terms  of  the  following:  Implication,  the  relation  of  a  term  to  the  class 
of  which  it  is  a  member,  the  notion  of  such  that,  the  notion  of  relation,  and 
such  further  notions  as  may  be  involved  in  the  general  notion  of  propositions 
of  the  above  form." 

The  content  of  mathematics,  on  this  view,  is  the  assertion  that  certain 
propositions  imply  certain  others,  and  these  propositions  are  all  expressible 
in  terms  of  "logical  constants",  that  is,  the  primitive  ideas  of  symbolic 
logic.  These  undefined  notions,  as  the  reader  is  already  aware,  need  not 
be  numerous :  ten  or  a  dozen  are  sufficient.  And  from  definitions  in  terms, 
finally,  of  these  and  from  the  postulates  of  symbolic  logic,  the  whole  of 
mathematics  is  deducible. 

The  logistic  development  of  a  mathematical  system  may,  like  the  arith- 
metic of  the  Formulaire,  assume  certain  undefined  mathematical  ideas  and 
mathematical  postulates  in  terms  of  these  ideas,  and  thus  differ  from  an 
ordinary  deductive  system  of  abstract  mathematics  only  by  expressing  the 
logical  ideas  which  occur  in  its  postulates  by  ideographic  symbols  and  by 
using  principles  of  proof  supplied  by  symbolic  logic.  Or  it  may,  like 
arithmetic  in  Principia  Mathematica,  assume  no  undefined  ideas  beyond 
those  of  logic,  define  all  its  mathematical  ideas  in  terms  of  these,  and  thus 
require  no  postulates  except,  again,  those  of  logic.  Or  it  may  pursue  an 

14  Principles  of  Mathematics,  p.  3. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  351 

intermediate  course,  assuming  some  of  its  ideas  as  primitive  but  defining 
others  in  terms  of  a  previously  developed  logic,  and  thus  require  some 
postulates  of  its  own  but  still  dispense  with  others  which  would  have  been 
necessary  in  a  non-logistic  treatment. 

But  whichever  of  these  modes  of  procedure  is  adopted,  the  general 
method  of  proof  in  logistic  will  be  the  same,  and  will  differ  from  any  non- 
logistic  treatment.  A  non-logistic  development  will  proceed  from  postulates 
to  theorems  by  immediate  inference  or  the  use  of  syllogism,  or  enthymeme, 
or  the  reductio  ad  absurdum,  and  such  general  logical  methods.  Or  it  may, 
upon  occasion,  make  use  of  methods  of  reasoning  the  validity  of  which 
depends  upon  the  subject  matter.  It  may  make  use  of  "mathematical 
induction",  which  requires  the  order  of  a  discrete  series  with  a  first  term. 
Or  if  proofs  of  consistency  and  independence  of  the  postulates  are  offered, 
these  will  make  use  of  logical  principles  which  are  most  complex  and  difficult 
of  comprehension — principles  of  which  no  thoroughly  satisfactory  account 
has  ever  been  given.  The  principles  of  all  this  reasoning  will  not  be  men- 
tioned; it  will  be  supposed  that  they  are  understood,  though  sometimes 
they  are  clear  neither  to  the  reader  nor  to  the  mathematician  who  uses 
them,  and  they  may  even  be  such  that  nobody  really  understands  them. 
(This  is  not  to  say  that  such  proofs  are  unsound.  Proofs  by  "mathematical 
induction  "  were  valid  before  Frege  and  Peano  showed  that  they  are  strictly 
deductive  in  all  respects.  But  in  mathematics  as  in  other  matters,  the 
assurance  or  recognition  of  validity  rests  upon  familiarity  and  upon  prag- 
matic sanctions  more  often  than  upon  consciously  formulated  principles.) 
As  contrasted  with  this,  the  logistic  method  requires  that  every  principle 
of  proof  be  explicitly  given,  because  these  principles  are  required  to  state 
each  step  of  proof. 

The  method  of  proof  in  logistic  is  sufficiently  illustrated  by  any  extended 
proof  of  Chapter  V.  Proofs  in  arithmetic  or  geometry  do  not  differ  in 
method  from  proofs  in  the  logic,  and  the  procedures  there  illustrated  are 
universal  in  logistic.  An  examination  of  these  proofs  will  show  that  postu- 
lates and  previously  established  theorems  are  used  as  principles  of  proof 
by  substituting  for  the  variables  p,  q,  r,  etc.,  in  these  propositions,  other 
expressions  which  can  be  regarded  as  values  of  their  variables.  The  general 
principle 

(P  -*  ?)  •*  (-?  •*  -P) 
can  thus  be  made  to  state 

(pq  +  p)-*  [-P  •*  -(P  q)] 


352  A  Survey  of  Symbolic  Logic 

by  substituting  p  q  for  p  and  p  for  q.  Or  if  /i  e  NC  be  substituted  for  p 
and  M  e  D'Nc  for  q,  it  states 

[(M  e  NC)  H  (M  e  D'Nc)]  -i  [-(/*  e  D'Xc)  ^  -(/*  e  NC)] 

Thus  any  special  case  which  comes  under  a  general  logical  principle  is 
stated  by  that  principle,  when  the  proper  substitutions  are  made.  This  is 
exactly  the  manner  in  which  the  principles  of  proof  which  belong  to  sym- 
bolic logic  state  the  various  steps  of  any  particular  proof  in  the  logistic 
development  of  arithmetic  or  geometry. 

Returning  to  our  first  example,  we  discover  that  in 

(p  q  -J  p)  -1 [-p  -J  -(p  q}] 

the  first  half,  p  q  •*  p,  is  itself  a  true  proposition.  Suppose  this  already 
proved  as,  in  fact,  it  is  in  the  last  chapter.  We  can  then  assert  what 
p  q  -J  p  is  stated  by  the  above  to  imply,  that  is, . 

-p  H  -(p  q) 

We  thus  prove  this  new  theorem  by  using  p  q  -J  p  as  a  premise.  To  use  a 
previous  proposition  as  a  premise  means,  in  the  logistic  method,  exactly 
this:  to  make  such  substitutions  in  a  general  principle  of  inference,  like 

(p  -*  q)  H  (-q  •»  -p) 

that  the  theorem  to  be  used  as  a  premise  appears  in  the  first  half  of  the 
expression— the  part  which  precedes  the  main  implication  sign.  That 
part  of  the  expression  which  follows  the  main  implication  sign  may  then 
be  asserted  as  a  consequence  of  this  premise. 

There  are  two  other  operations  which  may  be  used  in  the  proofs  of 
logistic — the  operation  of  substituting  one  of  a  pair  of  equivalent  expressions 
for  the  other,  and  the  operation  of  combining  two  previously  asserted 
propositions  into  a  single  assertion.15  The  first  of  these  is  exemplified 
whenever  we  make  use  of  a  definition.  For  example,  we  have,  in  the  system 
from  which  our  illustration  is  borrowed,  the  definition 

p  +  q  =  -(-p  -Q) 

and  the  theorem 

p  =  -(-p) 

15  The  operation  of  combining  two  propositions,  p  and  q,  into  the  single  assertion, 
p  q,  is  not  required  in  systems  based  on  material  implication,  because  we  have 

p  q  c.r  =  p  c  (q  cr) 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  353 

If  in  the  definition,  we  substitute  -p  for  p  and  -q  for  q,  it  states 

-p  +  -q  =  -[-(-p).-(-g)] 

And  then,  making  the  substitutions  which  the  theorem  p  =  -(-p)  allows, 
we  have 

-p  +  -q  =  -(p  q) 

which  may  be  asserted  as  a  theorem.     Again,  if  we  return  to  the  theorem 
proved  above, 

"  -p-i-(pg) 

we  are  allowed,  by  this  last  equivalence,  to  make  the  substitution  in  it  of 
-p  +  -q  for  -(p  q)'     Thus  we  prove 

-p  -i-p  +  -q 

This  sufficiently  illustrates  the  part  played  in  proof  by  the  substitution  of 
equivalent  expressions. 

We  may  now  see  exactly  what  the  mechanics  of  the  logistic  method  is. 
The  only  operations  required,  or  allowable,  in  proof  are  the  following: 

(1)  In  some  postulate  or  theorem  of  symbolic  logic,  other,  and  usually 
more  complex,  propositions  are  substituted  for  the  variables  p,  q,  r,  etc., 
which  represent  propositions.     The  postulate  or  theorem  in  which  these 
substitutions  are  made  is  thereby  used  as  a  principle  of  proof  which  states, 
in  this  particular  case,  the  proposition  which  results  when  these  substi- 
tutions are  made. 

(2)  The  postulate  or  theorem  of  logic  to  be  used  as  a  principle  of  proof 
may,  and  in  most  cases  does,  state  that  something  implies  something  else. 
In  that  event,  we  may  make  such  substitutions  as  will  produce  an  expression 
in  which  that  part  which  precedes  the  main  implication  sign  becomes 
identical  with  some  postulate  or  previously  proved  theorem — of  logic,  of 
arithmetic,  of  geometry,  or  whatever.     That  part  of  the  expression  which 
follows  the  main  implication  sign  may  then  be  separately  asserted  as  a  new 
theorem,  or  lemma,  which  is  thus  established.     The  postulate  or  previously 
proved  theorem  which  is  identical  with  what  precedes  the  main  implication 
sign,  in  such  a  case,  is  thus  used  as  a  premise. 

It  should  here  be  noted  that  propositions  of  logic,  of  geometry,  of  any 
logistic  system,  may  be  used  as  premises;  but  only  propositions  of  symbolic 
logic,  which  state  implications,  are  used  as  general  principles  of  inference. 

(3)  At  any  stage  of  a  demonstration,  one  of  a  pair  of  equivalent  expres- 
sions may  be  substituted  for  the  other. 

24 


354  A  Survey  of  Symbolic  Logic 

(4)  If,  for  example,  two  premises  are  required  for  a  certain  desired 
consequence,  and  each  of  these  premises  has  been  separately  proved,  then 
the  two  may  be  combined  in  a  single  assertion. 

These  are  all  the  operations  which  are  strictly  allowable  in  demonstra- 
tions by  the  logistic  method.  To  their  simplicity  and  definiteness  is 
attributable  a  large  part  of  the  precision  and  rigor  of  the  method.  Proof 
is  here  not  a  process  in  which  certain  premises  retire  into  somebody's  reason- 
ing faculty,  there  to  be  transformed  by  the  alchemy  of  thought  and  emerge 
in  the  form  of  the  conclusion.  The  whole  operation  takes  place  visibly  in 
the  successive  lines  of  work,  according  to  definite  rules  of  the  simplest 
possible  description.  The  process  is  as  infallible  and  as  mechanical  as  the 
adding  machine — except  in  the  choice  of  substitutions  to  be  made,  for  which, 
as  the  reader  may  discover  by  experiment,  a  certain  amount  of  intelligence 
is  required,  if  the  results  are  to  be  of  interest. 

III.    A  "HETERODOX"  VIEW  OF  THE  NATURE  OF  MATHEMATICS  AND  OF 

LOGISTIC 

We  have  now  surveyed  the  general  character  of  logistic  and  have  set 
forth  what  may  be  called  the  "orthodox"  view  of  it.  As  was  stated  earlier 
in  the  chapter,  the  account  which  has  now  been  given  is  such  as  would 
exclude  certain  systems  which  would  almost  certainly  be  classified  as 
logistic  in  their  character.  And  these  excluded  systems  are  most  naturally 
allied  with  another  view  of  logistic,  which  we  must  now  attempt  to  set 
forth.  The  differences  between  the  "orthodox"  and  this  "heterodox" 
view  have  to  do  principally  with  two  questions:  (1)  What  is  the  nature  of 
the  fundamental  operations  in  mathematics;  are  they  essentially  of  the 
nature  of  logical  inference  and  the  like,  or  are  they  fundamentally  arbitrary 
and  extra-logical  ?  (2)  Is  logistic  ideally  to  be  stated  so  that  all  its  assertions 
are  metaphysically  true,  or  is  its  principal  business  the  exhibition  of  logical 
types  of  order  without  reference  to  any  interpretation  or  application? 
The  two  questions  are  related.  -  It  will  appear  that  the  systems  which  the 
previous  account  of  logistic  did  not  cover  are  such  as  have  been  devised 
from  a  somewhat  different  point  of  departure.  One  might  characterize 
the  logistic  of  Principia  Mathematica  roughly  by  saying  that  the  order  of 
logic  is  assumed,  and  the  order  of  the  other  branches  then  follows  from  the 
meaning  of  their  terms.  On  the  other  hand,  the  systems  which  remain  to  be 
discussed  might,  equally  roughly,  be  characterized  by  saying  that  they 
attempt  to  set  up  a  type  of  logical  order,  which  shall  be  general  and  as 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  355 

inclusive  as  possible,  and  to  let  the  meaning  of  terms  depend  upon  their 
properties  of  order  and  relation.  Thus  this  "heterodox"  view  of  logistic 
is  one  which  takes  it  to  cover  all  investigations  and  developments  of  types 
of  logical  order  which  involve  none  but  ideographic  symbols  and  proceed 
by  operations  which  may  be  stated  with  precision  and  generality. 

In  any  case,  it  must  be  granted  that  the  operations  of  the  logistic  method 
are  themselves  pre-logical,  in  the  sense  that  they  underlie  the  proofs  of 
logic  as  well  as  of  other  branches.  The  assumption  of  these  operations — 
substitution,  etc. — is  the  most  fundamental  of  all  the  assumptions  of 
logistic.  It  is  possible  to  view  the  subject  in  a  way  which  makes  such 
pre-logical  principles  the  fundamentally  important  thing,  and  does  not 
regard  as  essential  the  use  of  symbolic  logic  as  a  foundation.  The  pro- 
priety of  the  term  logistic  for  such  studies  may  be  questioned.  But  if 
such  a  different  view  is  consistent  and  useful,  it  is  of  little  consequence 
what  the  method  ought  to  be  called. 

We  see  at  once  that,  if  such  a  view  can  be  maintained,  Mr.  Russell's 
definition  of  mathematics,  quoted  above,  is  arbitrary,  for  by  that  definition 
any  "logistic"  development  which  is  not  based  upon  logic  as  a  foundation 
will  not  be  mathematics  at  all.  As  a  fact,  it  will  be  simplest  to  present  this 
"heterodox"  view  of  logistic  by  first  presenting  and  explaining  the  cor- 
relative view  of  mathematics.  If  to  the  reader  we  seem  here  to  wander 
from  the  subject,  we  promise  to  return  later  and  draw  the  moral. 

A  mathematical  system  is  any  set  of  strings  of  recognizable  marks  in  which 
some  of  the  strings  are  taken  initially  and  the  remainder  derived  from  these 
by  operations  performed  according  to  rules  which  are  independent  of  any  mean- 
ing assigned  to  the  marks.  That  a  system  should  consist  of  marks  instead 
of  sounds  or  odors  is  immaterial,  but  it  is  convenient  to  discuss  mathe- 
matics as  written.  The  string-like  arrangement  is  due  simply  to  our  habits 
of  notation.  And  there  is  no  theoretical  reason  why  a  single  mark  may  not, 
in  some  cases,  be  recognized  as  a  "string". 

The  distinctive  feature  of  this  definition  lies  in  the  fact  that  it  regards 
mathematics  as  dealing,  not  with  certain  denoted  things — numbers,  tri- 
angles, etc. — nor  with  certain  symbolized  "concepts"  or  "meanings", 
but  solely  with  recognizable  marks,  and  dealing  with  them  in  such  wise 
that  it  is  wholly  independent  of  any  question  as  to  what  the  marks  repre- 
sent. This  might  be  called  the  "external  view  of  mathematics"  or  "mathe- 
matics without  meaning".  It  distinguishes  mathematics  from  other  sets 
of  marks  by  precisely  those  criteria  which  the  external  observer  can  always 


356  A  Survey  of  Symbolic  Logic 

apply.  Whatever  the  mathematician  has  in  his  mind  when  he  develops  a 
system,  what  he  does  is  to  set  down  certain  marks  and  proceed  to  manipulate 
them  in  ways  which  are  capable  of  the  above  description. 
^  This  view  is,  in  many  ways,  suggested  by  growing  tendencies  in  mathe- 
matics. Systems  become  "abstract",  entities  with  which  they  deal  "have 
no  properties  save  those  predicated  by  postulates  and  definitions",  and 
propositions  lose  their  phenomenal  reference.  It  becomes  recognized 
that  any  procedure  the  only  ground  for  which  lies  in  the  properties  of  the 
things  denoted — as  "constructions"  in  geometry — is  defective  and  un- 
mathematical.  Demonstrations  must  take  no  advantage  of  the  names 
by  which  the  entities  are  called.  But  if  Mr.  Russell  is  right,  the  mathe- 
matician has  given  over  the  metaphysics  of  space  and  of  the  infinite  only 
to  be  plunged  into  the  metaphysics  of  classes  and  of  functions.  Questions 
of  empirical  possibility  and  factual  existence  are  replaced  by  questions  of 
"logical"  possibility — questions  about  the  "existence"  of  classes,  about 
the  empty  or  null-class,  about  the  class  of  all  classes,  about  "individuals", 
about  "  descriptions  ",  about  the  relation  of  a  class  of  one  to  its  only  member, 
about  the  "values"  of  variables  and  the  "range  of  significance"  of  func- 
tions, about  material  and  formal  implication,  about  "types"  and  "system- 
atic ambiguities"  and  "hierarchies  of  propositions".  And  we  may  be 
pardoned  for  wondering  if  the  last  state  of  that  mathematician  is  not  worse 
than  the  first.  It  is  possible  to  think  that  these  logico-metaphysical 
questions  are  essentially  as  non-mathematical  as  the  earlier  ones  about 
empirical  possibility  and  phenomenal  existence.  One  may  maintain  that 
nothing  is  essential  in  a  mathematical  system  except  the  type  of  order. 
And  the  type  of  order  may  be  viewed  as  a  question  solely  of  the  distri- 
bution of  certain  marks  and  certain  complexes  of  marks  in  the  system. 
The  question  of  logical  meaning,  like  the  question  of  empirical  denotation, 
may  be  regarded  as  one  of  possible  applications  and  not  of  anything  internal 
to  the  system  itself. 

Before  discussing  the  matter  further,  it  may  prove  best  to  give  an  illus- 
tration. Let  us  choose  a  single  mathematical  system  and  see  what  we  shall 
make  of  it  by  regarding  it  simply  as  a  set  of  strings  of  marks. 

We  take  initially  the  following  eight  strings: 

(p*q)  =  (~p  v  ?) 

(P*q)  =  ~(~pv~q) 

(p  =  q)  =  ((p*q)  xfoDp)) 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  357 


We  must  now  state  rules  according  to  which  other  strings  can  be  derived 
from  the  above.  In  stating  these  rules,  we  shall  refer  to  quids  and  quods: 
these  words  are  to  have  no  connotation  ;  they  serve  merely  for  abbreviation 
in  referring  to  certain  marks. 

(1)  The  marks  +,    x,   D,  =  ,  and  =,  are  quods. 

(2)  The  marks  p,  q,  r,  are  quids;  and  any  recognizable  mark  not  appear- 
ing in  the  above  may  be  taken  arbitrarily  as  a  quid. 

(3)  Any  expression  consisting  of  two  quids,  one  quod,  and  the  marks 
)  and  (,  in  the  order  (quid  quod  quid),  may  be  treated  as  a  quid. 

(4)  The  combination  of  any  quid  preceded  immediately  by  the  mark  ~ 
may  be  treated,  as  a  quid. 

(5)  Any  string  in  the  set  may  be  repeated. 

(6)  Any  quid  which  is  separated  only  by  the  mark  =  from  some  other 
quid,  in  any  string  in  the  set,  may  be  substituted  for  that  other  quid  any- 
where. 

(7)  In  any  string  in  the  initial  set,  or  in  any  string  added  to  the  list 
according  to  rule,  any  quid  whatever  may  be  substituted  for  p  or  q  or  r, 
or  for  any  quid  consisting  of  only  one  mark.     When  a  quid  is  substituted 
for  any  mark  in  a  string,  the  same  quid  must  also  be  substituted  for  that 
same  mark  wherever  it  appears  in  the  string. 

(8)  The  string  resulting  from  the  substitution  of  a  quid  consisting  of 
more  than  one  mark  for  a  quid  of  one  mark,  according  to  (7),  may  be  added 
to  the  list  of  strings. 

(9)  In  any  string  added  to  the  list,  according  to  (8),  if  that  portion  of 
the  string  which  precedes  any  mark  2  is  identical  with  some  other  string 
in  the  set,  preceded  by  (,  then  the  portion  of  that  string  which  follows  the 
mark  D  referred  to  may  be  separately  repeated,  with  the  omission  of  the 
final  mark  ),  and  added  to  the  set. 

These  rules  are  unnecessarily  awkward.  In  the  illustration,  it  was 
important  not  to  refer  to  "propositions",  "relations",  "variables",  "paren- 
theses," etc.,  lest  it  should  not  be  clear  that  the  rules  are  independent  of 
the  meanings  of  the  marks.  But  though  cumbersome,  they  are  still  precise. 
The  original  eight  strings  of  marks  are,  with  minor  changes  of  notation, 


358  A  Survey  of  Symbolic  Logic 

definitions  and  postulates  of  divisions  #1  to  *5  in  Principia  Mathematica. 
By  following  the  rules  given,  anyone  may  derive  all  the  theorems  of  these 
divisions  and  all  other  consequences  of  these  assumptions,  without  knowing 
anything  about  symbolic  logic — either  before  or  after.  In  fact,  these 
rules  formulate  exactly  what  the  authors  have  done  in  proving  the  theorems 
from  the  postulates.16  For  this  reason,  it  is  unnecessary  to  carry  our  illus- 
tration further  and  actually  derive  other  strings  of  marks  from  the  initial 
set.  The  process  may  be  observed  in  detail  in  Principia  Mathematica: 
it  is,  in  all  important  respects,  the  same  with  the  process  of  proof  exhibited 
in  our  Chapter  V. 

The  method  of  development  in  Principia  Mathematica  differs  from  the 
one  we  have  suggested,  not  in  the  actual  manipulation  of  the  strings  of 
marks,  but  most  fundamentally  in  that  the  reasons  why — the  principles — 
of  their  operations  are  to  be  found,  not  in  explicitly  stated  rules,  but  in 
discussions  and  assumptions  concerning  the  conceptual  content  of  the 
system.  In  fact,  the  rules  of  operation  are  contained  in  explanations  of 
the  meaning  of  the  notation — in  discussions  of  the  nature  and  properties 
of  "elementary  propositions",  "elementary  prepositional  functions",  and 
so  forth.  For  example,  instead  of  stating  that  certain  substitutions  may 
be  made  for  p,  q,  r,  etc.,  they  assume  as  primitive  ideas  the  notions  of 
"elementary  propositions  "—p,  q,  r,  etc. — the  notion  of  "elementary 
prepositional  functions" — <px,  \l>z,  etc. — and  the  idea  of  "negation",  indi- 
cated by  writing  ~  immediately  before  the  proposition.  And  in  part,  the 
rules  of  operation  are  contained  in  certain  postulates,  distinguished  by 
their  non-symbolic  form:  "If  p  is  an  elementary  proposition,  ~p  is  an 
elementary  proposition",  "If  p  and  q  are  elementary  propositions,  pvq 
is  an  elementary  proposition",  and  "If  pp  and  $p  are  elementary  propo- 
sitional  functions  which  take  elementary  propositions  as  arguments,  <pp  v  \j/p 
is  an  elementary  prepositional  function.  The  warrant  for  the  substitution 
of  various  complexes  for  p,  q,  r,  etc.,  is  contained  in  these.  The  operation 
which  requires  our  complicated  rule  (9),  which  states  precisely  what  may 
be  done,  is  covered  by  their  assumptions:  "Anything  implied  by  a  true 
elementary  proposition  is  true",  and  "When  <px  can  be  asserted,  where  x 
is  a  real  variable,  and  <px  3  \f/x  can  be  asserted,  where  a:  is  a  real  variable, 
then  $x  can  be  asserted,  where  x  is  a  real  variable".  To  make  the  con- 
nection between  these  and  our  rule,  we  must  remember  that  3  is  the 

16  With  the  single  and  unimportant  exception  that  they  do  not  add  every  new  string 
which  they  arrive  at,  to  the  list  of  strings.  Many  such  are  simply  asserted  as  lemmas, 
used  immediately  for  one  further  proof  and  not  listed  as  theorems. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  359 

symbol  for  "implies",  that  if  what  precedes  this  sign  is  identical  with 
some  other  string  in  the  set,  that  means  that  what  precedes  is  true  or  is 
asserted,  and  that  the  number  of  '  open '  and  '  close '  parenthetic  marks  will 
indicate  whether  the  implication  in  question  is  the  main,  or  asserted,  impli- 
cation. 

We  have  chosen  this  particular  system  to  illustrate  the  requirements 
of  "mathematics  without  meaning"  for  a  special  reason,  which  will  appear 
shortly.  But  the  same  sort  of  modifications  would  be  sufficient  to  bring 
any  good  mathematical  system  into  this  form;  and  in  most  cases  such 
modifications  would  be  necessary. 

If,  for  example,  the  system  in  question  were  one  of  the  better-known 
algebras,  we  should  probably  have  "a  class,  K,  of  elements,  a,  b,  c,  etc.", 
and  such  assumptions  as  "  If  a  and  b  are  elements  in  K,  a  +  b  is  an  element 
in  K".  These  would  do  duty  as  the  principles  according  to  which,  for 
example,  x  +  y  would  be  substituted  for  a  in  any  symbolic  postulate  or 
theorem.  The  changes  in  such  a  system  would  be  less  radical,  hardly 
more  than  alterations  in  phraseology,  but  still  necessary. 

Reliance  upon  meanings  for  the  validity  of  the  method  has  obvious 
advantages.  It  is  simple  and  natural  and  clear.  (So  is  measuring  two 
line-segments  with  a  foot  rule  to  prove  equality.)  It  also  has  disadvantages. 
Besides  the  logico-metaphysical  questions  into  which  this  reliance  upon 
meanings  plunges  us,  there  is  the  disadvantage  that  it  works  a  certain 
confusion  of  the  form  of  the  system  with  its  content.  The  clear  separation 
of  these  is  the  ideal  set  by  "mathematics  without  meaning".  Not  only 
must  mathematical  procedure  be  free  from  all  appeal  to  intuition  or  to 
empirical  data;  it  should  also  be  independent  of  the  meaning  of  any  special 
concepts  which  constitute  the  subject  matter  of  the  system.  No  alteration 
or  abridgment  of  mathematical  procedure  anywhere  should  be  covered  by 
the  names  which  are  given  to  the  terms.  Only  those  relations  or  other 
properties  which  determine  a  system  as  a  particular  type  of  order  should 
be  allowed  to  make  a  difference  in  its  manner  of  development. 

To  secure  complete  separation  of  form  from  special  content,  and  to 
present  the  system  as  purely  formal  and  abstract,  means  precisely  to  use 
principles  of  operation  which  are  capable  of  statement  as  rules  for  the 
manipulation  of  marks — though,  in  general,  the  meticulous  avoidance  of 
any  reference  to  "meanings"  would  be  a  piece  of  pedantry.  The  important 
consideration  is  the  fact  that  the  operations  of  any  abstract  and  really  rigorous 
mathematical  system  are  capable  of  formulation  without  any  reference  to  truth 


360  A  Survey  of  Symbolic  Logic 

or  meanings.1'1  We  are  less  interested  in  any  superiority  of  this  "external 
view  of  mathematics",  or  in  the  conjectured  advantages  of  such  procedure 
as  has  been  suggested,  than  in  its  bare  possibility.  If  the  considerations  here 
presented  are  not  wholly  mistaken,  then  the  ideal  of  form  which  requires 

17  It  is  possible  to  regard  such  manipulation  of  marks,  the  discovery  of  sufficiently 
precise  rules  and  of  initial  strings  which  will,  together,  determine  certain  results,  and  the 
exhibition  of  the  results  which  such  systems  give,  as  the  sole  business  of  the  mathematician. 

Mathematics,  so  developed,  achieves  the  utmost  economy  of  assertion.  Nothing  is 
asserted.  There  are  no  primitive  ideas.  Since  no  meanings  are  given  to  the  characters, 
the  strings  are  neither  true  nor  false.  Nothing  is  assumed  to  be  true,  and  nothing  is  asserted 
as  "proved".  It  is  not  even  necessary  to  assert  that  certain  operations  upon  certain  marks 
give  certain  other  marks.  The  initial  strings  are  set  down:  the  requirements  of  pure 
mathematics  are  satisfied  if  the  others  are  got  and  recorded.  Yet  these  initial  strings  and 
the  rules  of  operation  determine  a  definite  set  of  strings  of  marks — determine  unambiguously 
and  absolutely  a  certain  mathematical  system. 

To  many,  such  a  view  will  seem  to  exclude  from  mathematics  everything  worthy  of 
the  name.  These  will  urge  that  the  modern  developments  of  mathematics  have  aimed 
at  exact  analysis  into  fundamental  concepts;  that  this  analysis  does,  as  a  fact,  bring  about 
such  simplification  of  the  essential  operations  as  to  make  possible  mechanical  manipulation 
of  the  system  without  reference  to  meanings;  but  that  it  is  absurd  to  take  this  shell  of 
refined  symbolism  for  the  meat  of  mathematics.  To  any  such,  it  might  be  replied  that 
the  development  of  kinematics  as  an  abstract  mathematical  system  does  not  remove  the 
physics  of  matter  in  motion  from  the  field  of  experimental  investigation;  that  abstract 
geometry  still  leaves  room  for  all  sorts  of  interesting  inquiry  about  the  nature  of  our  space : 
that  for  every  system  which  is  freed  from  empirical  denotations  there  is  created  the  separate 
investigation  of  the  possible  applications  of  this  system.  Correspondingly,  for  every 
system  which  is  made  independent  of  classes,  individuals,  relations,  and  so  on,  there  is 
created  the  separate  investigation  of  the  metaphysical  status  of  the  classes,  individuals, 
and  relations  in  question — of  the  application  of  the  system  of  marks  to  systems  of  more 
special  "concepts",  i.  e.  to  systems  of  logical  and  metaphysical  entities.  That  we  are 
more  interested  in  the  applications  of  a  system  than  in  its  rigorous  development,  more 
interested  in  its  "meaning"  than  in  its  structure,  should  not  lead  to  a  confusion  of  meaning 
with  structure,  of  applications  with  method  of  development. 

It  may  be  further  objected  that  this  view  seems  to  remove  mathematics  from  the 
field  of  science  altogether  and  make  it  simply  an  art;  that  the  computer  would,  by  this 
definition,  be  the  ideal  mathematician.  But  there  is  one  feature  of  mathematics,  even  as  a 
system  of  marks,  which  is  not,  and  cannot  be  made,  mechanical.  Valid  results  may  be 
ob tamed  by  mechanical  operations,  and  each  single  step  may  be  essentially  mechanical, 
yet  the  derivation  of  "required"  or  "interesting"  or  "valuable"  results  will  need  an  in- 
telligent and  ingenious  manipulator.  Gulliver  found  the  people  of  Brobdingnag  (?)  feed- 
ing letters  into  a  machine  and  waiting  for  it  to  turn  out  a  masterpiece.  Well,  master- 
pieces are  combinations  achieved  by  placing  letters  in  a  certain  order!  However  mechanical 
the  single  operation,  it  will  take  a  mathematician  to  produce  masterpieces  of  mathematics. 
A  machine,  or  machine-like  process,  will  start  from  something  given,  take  steps  of  a  deter- 
mined nature,  and  render  the  result,  whatever  it  is;  but  it  will  not  choose  its  point  of 
departure  and  select,  out  of  various  possibilities,  the  steps  to  be  taken  in  order  to  achieve  a 
desired  result.  Is  not  just  this  ingenuity  in  controlling  the  destination  of  simple  operations 
the  peculiar  skill  which  mathematics  requires?  The  mathematician,  like  any  other  sci- 
entific investigator,  is  largely  engaged  upon  what  are,  from  the  point  of  view  of  the  finished 
science,  inverse  processes:  he  gets,  by  trial  and  error,  or  intuition,  or  analogy,  what  he 
presents  finally  as  rigidly  necessary.  To  produce  or  reveal  necessities  previously  un- 
noticed— this  is  the  peculiar  artistry  of  his  work. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  361 

that  mathematics  abstract  not  only  from  possible  empirical  meanings  but 
also  from  logical  or  metaphysical  meanings  is  a  wholly  attainable  ideal. 
And  if  this  is  possible,  then  Mr.  Russell's  view  that  "Pure  Mathematics 
is  the  class  of  all  propositions  of  the  form  'p  implies  q',  etc.",  is  an  arbitrary 
definition,  and  the  ideal  of  form  which  it  imposes  is  not  a  necessary  one, 
but  must  take  its  chances  with  other  such  ideals.  The  decision  among 
these  will,  then,  be  a  matter  of  choice,  dependent  upon  the  advantages  to 
be  gained  by  one  or  the  other  form.  There  is  no  a  priori  reason  why 
systems  which  are  generated  by  "mathematical"  operations,  some  of  which 
may  be  peculiar  to  the  system  and  meaningless  in  logic,  are  not  just  as 
"sound"  and  "good"  and  even  "ideal"  as  systems  developed  by  the  com- 
pletely analytical  method  of  Principia  Mathematica  which  reduces  all 
operations  to  those  of  logic.  And  "extra-logical"  modes  of  development 
may  be  just  as  universal  as  the  "logical",  since  symbolic  logic  itself  may  be 
developed  by  the  "extra-logical"  method.  It  was  to  make  this  clear  that 
we  chose  the  particular  system  which  we  did  for  our  illustration. 

In  fact,  symbolic  logic,  or  that  branch  of  it  which  is  developed  first  as  a 
basis  for  others,  must  be  developed  by  operations  the  validity  of  which  is 
presumed  apart  from  the  logic  so  developed.  It  may,  indeed,  be  the  case 
that  logic  is  developed  by  methods  which  it  validates  by  its  own  theorems, 
when  these  are  proved;  it  may  thus  be  "self-critical",  or  "circular"  in  a 
sense  which  means  consistency  rather  than  fallacy.  But  this  is  not  really 
to  the  point:  if  the  validity  of  certain  operations  is  presupposed,  then  that 
validity  is  presupposed,  whether  it  is  afterward  proved  valid  or  not.  There 
is,  then,  a  certain  advantage  in  the  explicit  recognition  that  a  system  of 
symbolic  logic  is  merely  a  set  of  strings  of  marks,  manipulated  by  certain 
arbitrary  and  "extra-logical"  principles.  It  is,  in  fact,  only  on  this  view 
that  symbolic  logic  can  be  abstract.  For  symbolic  logic,  as  has  already 
been  pointed  out,  is  peculiar  among  mathematical  systems  in  that  its  postu- 
lates and  theorems  are  used  to  state  proofs.  If,  then,  the  proofs  are  to  be 
logically  valid,  these  postulates  and  theorems  must  be  true,  and  the  system 
cannot  be  abstract.  But  if  the  "proofs"  are  required  to  be  "valid"  only 
in  the  sense  that  certain  arbitrary  and  extra-logical  rules  for  manipulation 
have  been  observed,  then  it  matters  no  more  in  logic  than  in  any  other 
branch  whether  the  propositions  be  true,  or  even  what  they  mean.  There 
is  the  same  possibility  of  choice  here  that  there  is  in  the  case  of  other  mathe- 
matical systems — the  choice  which  is  phrased  most  sharply  as  the  alterna- 
tive between  the  Russellian  view  and  the  "external  view  of  mathematics".18 

18  It  may  be  noted  that  if  mathematics  consists  of  "propositions  of  the  form  'p  im- 


362  A  Survey  of  Symbolic  Logic 

If  we  take  this  view  of  mathematics,  or  any  view  which  regards  arbitrary 
mathematical  operations  as  equally  fundamental  with  the  operations  of 
logic,  we  shall  then  give  a  different  account  of  logistic  and  of  its  relation 
to  logic.  We  shall,  in  that  case,  regard  symbolic  logic  as  one  mathematical 
system,  or  type  of  order,  among  others.  We  shall  recognize  the  possibility 
of  generating  all  other  types  of  order  from  the  order  of  logic,  but  we  shall 
find  no  necessity  in  this  proceeding.  We  may,  possibly,  find  some  other 
very  general  type  of  order  from  which  the  order  of  logic  may  be  derived. 
And  the  question  of  any  hierarchic  arrangement  of  systems  will  then  depend 
upon  convenience  or  simplicity  or  some  other  pragmatic  consideration. 
Logistic  will,  then,  be  defined  not  by  any  relation  to  symbolic  logic  but  as 
the  study  of  types  of  order  as  such,  or  as  any  development  of  mathematics 
which  seeks  a  high  degree  of  generality  and  complete  independence  of 
any  particular  subject  matter. 

IV.    THE  LOGISTIC  METHOD  or  KEMPE  AND  ROYCE 

We  should  not  care  to  insist  upon  the  "external  view  of  mathematics" 
and  the  consequent  view  of  logistic  which  has  been  outlined.  Other  con- 
siderations aside,  it  seems  especially  dubious  to  dogmatize  about  the  ideal 
of  mathematical  form  when  there  is  no  common  agreement  on  the  topic 
among  mathematicians.  But  we  can  now  answer  the  questions  which 
prefaced  this  discussion:  Are  the  fundamental  operations  of  mathematics 
those  of  logic  or  are  they  extra-logical?  And  is  logistic  ideally  to  be  so 
stated  that  all  its  assertions  are  metaphysically  true  or  is  it  concerned  simply 
to  exhibit  certain  general  types  of  order?  The  answer  is  that  it  is  entirely  a 
matter  of  choice,  since  either  view  can  consistently  be  maintained  and 
mathematics  be  developed  in  the  light  of  it.19  This  is  especially  important 
for  us,  since,  as  has  been  mentioned,  there  are  certain  studies  which  would 
most  naturally  be  called  logistic  which  would  not  be  covered  by  the  "  ortho- 
plies  q',  where  p  and  q  are  propositions  containing  one  or  more  variables,  etc.",  and  the 
theorems  about  "implies"  are  required  to  be  true  if  proofs  are  to  be  valid,  all  mathematics 
must  be  true  in  order  to  be  valid.  On  this  view,  "  abstractness "  can  reside  only  in  the 
range  of  the  variables  contained  in  p  and  q. 

19  One  case  in  which  the  "external  view  of  mathematics"  is  highly  convenient,  is  of 
especial  interest  to  us.  There  are  various  symbolic  "logics"  which  differ  from  one  another 
both  in  method  and  in  content.  Discussion  of  the  correctness  and  relative  values  of  these 
is  almost  impossible  unless  we  recognize  that  the  order  of  logic  can  be  viewed  quite  apart 
from  its  content — that  a  symbolic  logic  may  be  abstract,  just  like  any  other  branch  of 
mathematics — and  thus  separate  the  question  of  mathematical  consistency  (of  mere  ob- 
servance of  arbitrary  and  precise  principles  of  operation)  from  questions  of  applicability 
of  a  system  to  valid  reasoning.  The  difficulty  of  making  this  separation  hampered  our 
discussion  of  "implies"  in  the  last  chapter. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  363 

dox"  view,  since  they  are  based,  not  upon  logic,  but  upon  an  order  prior 
to  or  inclusive  of  logic.  These  studies  exemplify  a  method  which  differs 
in  notable  respects  both  from  that  of  Peano  and  that  of  Principia  Mathe- 
matica.  And  it  seems  highly  desirable  that  we  should  discuss  this  alterna- 
tive method  without  initial  prejudice. 

It  is  characteristic  of  this  alternative  method  that  it  seeks  to  define 
initially  a  field,  or  class  of  entities,  and  an  order  in  this  field,  which  shall  be 
mathematically  as  inclusive  as  possible,  so  that  more  special  orders  may  be 
specified  by  principles  of  selection  amongst  the  entities.  It  is  distinguished 
from  the  method  followed  by  Peano  in  the  Formulaire  by  the  fact  that  it 
seeks  to  get  special  orders,  such  as  that  of  geometry,  without  further 
"existence  postulates",  and  from  the  method  of  Principia  Mathematica  by 
the  attempt  to  substitute  selection  within  an  initial  order  for  analysis  (defini- 
tion by  previous  ideas)  of  newly  introduced  terms.  The  result  is  that  this 
method  is  particularly  adapted  to  exhibit  the  analogies  of  different  special 
fields — the  partial  identities  of  various  types  of  order. 

The  application  of  this  method  has  not  been  carried  out  extensively 
enough  so  that  we  may  feel  certain  either  of  its  advantages  or  of  its  limita- 
tions. The  method  is,  in  a  certain  sense,  exemplified  wherever  we  have 
various  mathematical  systems  all  of  which  satisfy  a  given  set  of  postulates, 
but  each— or,  say,  all  but  one — satisfying  some  one  or  more  of  the  postulates 
"vacuously".  For  here  we  have  an  ordered  field  within  which  other  and 
more  limited  systems  are  specified  by  a  sort  of  selection.  ("Selection"  is 
not  the  proper  word,  but  no  better  one  has  occurred  to  us.)  It  is  particu- 
larly in  two  studies  of  the  relation  of  geometry  to  logic  that  the  method  has 
been  consciously  followed: 20  in  a  paper  by  A.  B.  Kempe,  "On  the  Relation 
between  the  Logical  Theory  of  Classes  and  the  Geometrical  Theory  of 
Points,"21  and  in  Josiah  Royce's  study,  "The  Relation  of  the  Princi- 
ples of  Logic  to  the  Foundations  of  Geometry".22  We  shall  hardly  wish  to 
go  into  these  studies  in  detail,  but  something  of  the  mode  of  procedure  and 
general  character  of  the  results  achieved  may  be  indicated  briefly. 

Kempe  enunciates  the  principle  that  "...  so  far  as  processes  of  exact 
thought  are  concerned,  the  properties  of  any  subject  matter  depend  solely 
on  the  fact  that  it  possesses  'form' — i.  e.,  that  it  consists  of  a  number  of 

20  Peirce's  system  of    "logical   quaternions"   (see  above,  pp.  102-04)    also  exhibits 
something  of  this  method. 

21  Proc.  London  Math.  Soc.,  xxi  (1890),  147-82. 

22  Trans.  Amer.  Math.  Soc.,  vi  (1905),  353^15. 

Some  portions  of  the  discussion  of  this  paper  and  Kempe's  are  here  reprinted  from  an 
article,  "Types  of  Order  and  the  System  S, "  in  Phil.  Rev.  for  May,  1916. 


364  A  Survey  of  Symbolic  Logic 

entities,  certain  individuals,  pairs,  triads,  &c.,  certain  of  which  are  exactly  like 
each  other  in  all  their  relations,  and  certain  not;  these  like  and  unlike  indi- 
viduals, pairs,  triads,  &c.,  being  distributed  through  the  whole  system  of 
entities  in  a  definite  way".23  In  illustration  of  this  theory,  he  seeks  to 
derive  the  order  both  of  logical  classes  and  of  geometrical  sets  of  points 
from  assumptions  in  terms  of  a  triadic  relation,  a  c  •  b,  which  may  be  read 
"b  is  'between'  a  and  c".  The  type  of  this  relation  may  be  illustrated  as 
follows :  Let  a,  b,  c  represent  areas ;  then  a  c  •  b  symbolizes  the  fact  that  b 
includes  whatever  area  is  common  to  a  and  c,  and  is  itself  included  in  the 
area  which  comprises  what  is  either  a  or  c  or  common  to  both.  Or  it 
may  be  expressed  in  the  Boole-Schroder  Algebra  as 

(a  c)  c  b  c  (a  +  c),         or        a  -b  c  +  -a  b  -c  =  0 

The  essential  properties  of  serial  order  may  be  formulated  in  terms  of  this 
relation.  If  ac-b  and  ad-c,  then  also  ad-b  and  bd-c.  If  b  is  between 
a  and  c  and  c  is  between  a  and  d,  then  also  b  is  between  a  and  d,  and  c  is 
between  b  and  d.24 

abed 

Thus  the  relation  gives  the  most  fundamental  property  of  linear  sets. 
If  a  be  regarded  as  the  origin  with  reference  to  which  precedence  is  deter- 

23  "On  the  Relation  between,  etc.",  loc.  cit.,  p.  147. 

24  Assuming  ac-b  to  be  expressed  in  the  Boole-Schroder  Algebra  (as  above)  by 

(a  c)  c  b  c  (a  +  c) 
this  deduction  is  as  follows: 

ac-b  is  equivalent  to  (a  c)  c  b  c  (a  +  c). 
ad  -c  is  equivalent  to  (ad)  c  c  c  (a  +  d). 

By  the  laws  of  the  algebra, 
(a  c)  c  b  is  equivalent  to  a  -b  c  =0. 
b  c  (a  +  c)  is  equivalnt  to  -(a  +  c)  b  =  -a  b  -c  =  0. 
(a  d)  c  c  is  equivalent  to  a  -c  d  =0. 
c  c  (a  +  d)  is  equivalent  to  -(a  +  d)  c  =  -a  c  -d  =  0. 

Combining  these  premises,  i.  e.  adding  the  equations,  we  have 
a  -b  c  +  -a  b  -c  +  a  -c  d  +  -a  c  -d  =  0. 

Expanding  each  term  of  the  left-hand  member  with  reference  to  that  one  of  the 
elements,  a,  b,  c,  d,  not  already  involved  in  it, 
a  -b  c  d  +  a  -b  c  -d  +  -a  b  -c  d  +  -a  b  -c  -d  +  a  b  -c  d  +  a  -b  -c  d 

+  -a  b  c  -d  +  -a  -b  c  -d  =  0 

By  the  law  "If  a  +  b  =  0,  a  =  0",  we  get  from  this, 

(1)  a  -b  d  (c  +  -c)  +  -a  b  -d  (c  +  -c)  =0  =  a  -b  d  +  -a  b  -d, 
and  (2)  b  -c  d  (a  +  -a)  +  -b  c  -d  (a  +  -a)  =  0  =  b  -c  d  +  -b  c  -d. 

(1)  is  equivalent  to  (a  d)  c  b  c  (a  +  d),  or  ad-b, 
and  (2)  is  equivalent  to  (6  d)  c  c  c  (b  +  d),  or  bd-c. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  365 

mined,  ac-b  will  represent  "b  precedes  c",  and  a  d-c  that  "c  precedes  d". 
Since  ac-b  and  a  d-c  together  give  ad-b,  we  have:  if  "b  precedes  c"  and 
"c  precedes  d",  then  "b  precedes  d".  Hence  this  relation  has  the  essential 
transitivity  of  serial  order,  with  the  added  precision  that  it  retains  reference 
to  the  origin  from  which  "precedes"  is  determined. 

The  last-mentioned  property  of  this  relation  makes  possible  an  inter- 
pretation of  it  for  logical  classes  in  which  it  becomes  more  general  than 
the  inclusion  relation  of  ordinary  syllogistic  reasoning.  If  there  should 
be  inhabitants  of  Mars  whose  logical  sense  coincided  with  our  own — so 
that  any  conclusion  which  we  regarded  as  valid  would  seem  valid  to  them, 
and  vice  versa — but  whose  psychology  was  somewhat  different  from  ours, 
these  Martians  might  prefer  to  remark  that  "b  is  'between'  a  and  c", 
rather  than  to  note  that  "All  a  is  b  and  all  b  is  c".  These  Martians  might 
then  carry  on  successfully  all  their  reasoning  in  terms  of  this  triadic  '  between' 
relation.  For  ac-b,  meaning 

-a  b  -c  +  a  -b  c  =  0 

is  a  general  relation  which,  in  the  special  case  where  a  is  the  "null"  class 
contained  in  every  class,  becomes  the  familiar  "b  is  contained  in  c"  or 
"All  b  is  c  ".  By  virtue  of  the  transitivity  pointed  out  above,  0  c  -  b  and  0  d-c 
together  give  0  d-b,  which  is  the  syllogism  in  Barbara,  "If  all  b  is  c  and  all 
c  is  d,  then  all  b  is  d".  Hence  these  Martians  would  possess  a  mode  of 
reasoning  more  comprehensive  than  our  own  and  including  our  own  as  a 
special  case. 

The  triadic  relation  of  Kempe  is,  then,  a  very  powerful  one,  and  capable 
of  representing  the  most  fundamental  relations  not  only  in  logic  but  in 
all  those  departments  of  our  systematic  thinking  where  unsymmetrical 
transitive  (serial)  relations  are  important.25  In  terms  of  these  triads, 
Kempe  states  the  properties  of  his  "base  system",  from  whose  order  the 
relations  of  logic  and  geometry  both  are  derived.  The  "base  system" 
consists  of  an  infinite  number  of  homogeneous  elements,  each  having  an 
infinite  number  of  equivalents.  It  is  assumed  that  triads  are  disposed  in 
this  system  according  to  the  following  laws : 26 

26  It  should  be  pointed  out  that  while  capable  of  expressing  such  relations,  this  triadic 
relation  is  itself  not  necessarily  unsymmetrical:  ac-b  and  ab-c  may  both  be  true.  But  in 
that  case,  b  =  c,  as  may  be  verified  by  adding  the  equations  for  these  two  triads.  Further, 
for  any  a  and  b,  ab-a  and  ab-b  always  hold — &  is  always  contained  in  itself.  Thus  the 
triadic  relation  represents  serial  order  with  the  qualification  that  any  term  "precedes" 
itself  or  is  "between"  itself  and  any  other — an  entirely  intelligible  and  even  useful  con- 
vention. 

26  See  Kempe's  paper,  loc.  tit.,  pp.  148-49. 


366  A  Survey  of  Symbolic  Logic 

1.  If  we  have  a  b-p  and  cb-q,  r  exists  such  that  we  have  a  q-r  and  c  p-r. 

2.  If  we  have  a  b •  p  and  c  p-r,  q  exists  such  that  we  have  a  q •  r  and  c  b •  r.27 

3.  If  we  have  a  b  •  c  and  a  =  b,  then  c  =  a  =  b. 

4.  If  a  =  b,  then  we  have  a  c  •  b  and  b  c  •  a,  whatever  entity  of  the  system 
c  may  be. 

To  these,  Kempe  adds  a  fifth  postulate  which  he  calls  "the  Law  of  Con- 
tinuity: "No  entity  is  absent  from  the  system  which  can  consistently  be 
present".  From  these  assumptions  and  various  definitions  in  terms  of 
the  triadic  relation,  he  is  able  to  derive  the  laws  of  the  symbolic  logic  of 
classes  and  fundamental  properties  of  geometrical  sets  of  points.  But 
further  and  most  important  properties  of  geometrical  sets  depend  upon  the 
selection  of  such  sets  within  the  "base  system"  by  the  law: 28 

If  we  have  a-p-q, 
b-p-q, 

and        a-b-p    does  not  hold; 
then       p  =  q. 

'a-p-q'  here  represents  a  relation  of  a,  p,  and  q,  such  that  some  one  at 
least  of  ap-q,  aq-p,  pq-a  will  hold.  If  we  call  ab-c  a  "linear  triad", 
then  the  set  or  locus  selected  by  the  above  law  will  be  such  that  no  two 
linear  triads  of  the  'points'  comprised  in  it  can  have  two  non-equivalent 
'points'  in  common.  Of  such  a  geometric  set,  Kempe  says:29  "It  is 
precisely  the  set  of  entities  which  is  under  consideration  by  the  geometrician 
when  he  is  considering  the  system  of  points  which  make  up  flat  space  of 
unlimited  dimensions". 

But  there  are  certain  dubious  features  of  Kempe 's  procedure.  As  Pro- 
fessor Royce  notes,  the  Law  of  Continuity  makes  postulates  1  and  2  super- 
fluous. And  there  are  other  objections  to  it  also.  Moreover,  in  spite  of 
the  fact  that  Kempe  has  assumed  an  infinity  of  elements  in  the  "base  set", 
there  are  certain  ambiguities  and  difficulties  about  the  application  of  his 
principles  to  infinite  collections. 

In  Professor  Royce's  paper,  we  have  no  such  'blanket  assumption'  as 
the  Law  of  Continuity,  and  the  relations  defined  may  be  extended  without 
difficulty  to  infinite  sets.  We  have  here,  in  place  of  the  "base  system" 
and  triadic  relations,  the  "system  S",  the  "^-relation"  and  the  "0-rela- 

27  If  the  reader  will  draw  the  triangle,  a  be,  and  put  in  the  "  betweens"  as  indicated, 
the  geometrical  significance  of  these  postulates  will  be  evident.     I  have  changed  a  little 
the  order  of  Kempe's  terms  so  that  both  1  and  2  will  be  illustrated  by  the  same  triangle. 

28  See  Kempe's  paper,  loc.  cit.,  pp.  176-77. 

29  Ibid.,  p.  177. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  367 

tion".  The  F-relation  is  a  polyadic  relation  such  that  F(ab. .  ./xy.  .  .)  is 
expressible  in  the  Boole-Schroder  Algebra  as 

a  b  ... -x  -y  .  .  .  +  -a  -b  ... .  x  y  .  .  .  =  0 

This  is  the  generalization  of  Kempe's  a  c-b,  which  is  F(ac/b).  The  0-rela- 
tion  is  a  polyadic  symmetrical  relation  which  expresses  simultaneously  a 
whole  set  of  equivalent  ^-relations.  0(abc.  .  .)  is  expressible  as 

a  b  c  .  .  .  +  -a  -b  -c  .  .  .   =  0 

We  have  used  the  algebra  of  classes  to  express  these  relations,  but  in 
Professor  Royce's  paper,  this  order  is,  of  course,  reversed.  In  terms  of  0- 
relations,  the  ideas  of  the  logic  of  classes  are  defined,  and  from  the  postula- 
tion  of  certain  0-relations,  the  laws  of  the  symbolic  logic  of  classes  are  de- 
rived. And,  in  most  interesting  ways  which  we  cannot  here  discuss,  the 
order  of  the  system  S  is  also  shown  to  possess  all  the  fundamental  proper- 
ties of  geometric  sets  of  points.  The  system  S  has  a  structure  such  that  it 
might  be  called  "the  logical  continuum",  and  there  are  good  grounds  for 
presuming  that  types  of  order  in  the  greatest  variety  may  be  specified  within 
the  system  simply  by  selection.  In  the  words  of  Professor  Royce: 30 

"Wherever  a  linear  series  is  in  question,  wherever  an  origin  of  coordi- 
nates is  employed,  wherever  'cause  and  effect',  'ground  and  consequence/ 
orientation  in  space  or  direction  of  tendency  in  time  are  in  question,  the 
diadic  asymmetrical  relations  involved  are  essentially  the  same  as  the  rela- 
tion here  symbolized  by  p  —<  yq,  [' q  is  "between"  y  and  p';  or,  with  y 
as  origin,  'p  precedes  q';  or,  where  y  is  the  null-class,  'p  is  contained 
in  q';  or,  in  terms  of  propositions,  'p  implies  q'].  This  expression,  then, 
is  due  to  certain  of  our  best  established  practical  instincts  and  to  some  of 
our  best  fixed  intellectual  habits.  Yet  it  is  not  the  only  expression  for  the 
relations  involved.  It  is  in  several  respects  inferior  to  the  more  direct 
expression  in  terms  of  0-relations.  .  .  .  WThen,  in  fact,  we  attempt  to  de- 
scribe the  relations  of  the  system  2  merely  in  terms  of  the  antecedent- 
consequent  relation,  we  not  only  limit  ourselves  to  an  arbitrary  choice  of 
origin  [y  in  p  —<  „  q],  but  miss  the  power  to  survey  at  a  glance  relations 
of  more  than  a  diadic,  or  triadic  character." 

V.    SUMMARY  AND  CONCLUSION 

There  are,  then,  in  general,  three  types  of  logistic  procedure.     There  is, 
first,  the  "simple  logistic  method",  as  we  may  call  it — the  most  obvious 
30  "The  Relations  of  the  Principles  of  Logic,  etc.,"  loc.  cit.,  pp.  381-82. 


368  A  Survey  of  Symbolic  Logic 

one,  in  which  the  various  branches  of  pure  mathematics,  taken  in  the  non- 
logistic  but  abstract  form,  are  simply  translated  into  the  logistic  terms  which 
symbolize  ideographically  the  relations  involved  in  proof.  When  this 
translation  is  made,  the  proofs  in  arithmetic,  or  geometry,  etc.,  will  be 
simply  special  cases  of  the  propositions  of  symbolic  logic.  But  other 
branches  than  logic  will  have  their  own  primitive  or  undefined  ideas  and 
their  own  postulates  in  terms  of  these.  We  have  used  Peano's  Formulaire 
as  an  illustration  of  this  method,  although  the  Formulaire  has,  to  an  extent, 
the  characters  of  the  procedure  to  be  mentioned  next.  Second,  there  is 
the  hierarchic  method,  or  the  method  of  complete  analysis,  exemplified  by 
Principia  Mathematica.  Here  the  calculus  of  propositions  (or  implications) 
is  first  developed,  because  by  its  postulates  and  theorems  all  the  proofs  of 
other  branches  are  to  be  stated.  And,  further,  all  the  terms  and  relations 
of  other  branches  are  to  be  so  analyzed,  i.  e.,  defined,  that  from  their  defini- 
tion and  the  propositions  of  the  logic  alone,  without  additional  primitive 
ideas  or  postulates,  all  the  properties  of  these  terms  will  follow.  And, 
third,  there  is  the  method  of  Kempe  and  Royce.  This  method  aims  to 
generate  initially  an  order  which  is  not  only  general,  as  is  the  order  of  logic, 
but  inclusive,  so  that  the  type  of  order  of  various  special  fields  (in  as  large 
number  and  variety  as  possible)  may  be  derived  simply  by  selection — i.  e., 
by  postulates  wrhich  determine  the  class  which  exhibits  this  special  order  as 
a  selection  of  members  of  the  initially  ordered  field.31  For  this  third  method, 
other  types  of  order  will  not  necessarily  be  based  upon  the  order  of  logic: 
in  the  only  good  examples  which  we  have  of  the  method,  logic  is  itself 
derived  from  a  more  inclusive  order.  The  sense  in  which  such  a  procedure 
may  still  be  regarded  as  logistic  has  been  made  clear  in  what  precedes. 

Which  of  these  methods  will,  in  the  end,  prove  most  powerful,  no  one 
can  say  at  present.  The  whole  subject  of  logistic  is  too  new  and  un- 
developed. But  certain  characters  of  each,  indicating  their  adaptability, 
or  the  lack  of  it,  to  certain  ends,  can  be  pointed  out.  The  hierarchic  or 
completely  analytic  method  has  a  certain  imposing  quality  which  right- 
fully commands  attention.  One  feels  that  here,  for  once,  we  have  got  to 
the  bottom  of  things.  Any  work  in  which  this  method  is  extensively  carried 
out,  as  it  is  in  Principia  Mathematica,  is  certainly  monumental.  Further, 
the  method  has  the  advantage  of  setting  forth  various  branches  of  the 
subject  investigated  in  the  order  of  their  logical  simplicity.  And  the  step 

31  Professor  Royce  used  to  say  facetiously  that  the  system  S  had  some  of  the  properties 
of  a  junk  heap  or  a  New  England  attic.  Almost  everything  might  be  found  in  it:  the  ques- 
tion was,  how  to  get  these  things  out. 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  369 

from  one  such  division  to  another  based  upon  it  is  always  such  as  to  make 
clear  the  connection  between  the  two.  The  initial  analyses — definitions — 
which  make  such  steps  possible  are,  indeed,  likely  to  tax  our  powers,  but 
once  the  initial  analysis  is  correctly  performed,  the  theorems  concerning  the 
derived  order  will  be  demonstrable  by  processes  which  have  already  become 
familiar  and  even  stereotyped.  The  great  disadvantage  of  this  completely 
analytic  method  is  its  great  complexity  and  the  consequent  tediousness  of 
its  application.  It  is  fairly  discouraging  to  realize  that  the  properties  of 
cardinal  number  require  some  four  hundred  pages  of  prolegomena — in  a 
symbolism  of  great  compactness — for  their  demonstration.  To  those  whose 
interests  are  simply  "mathematical"  or  "scientific"  in  the  ordinary  sense, 
it  is  forbidding. 

The  simple  logistic  method  offers  an  obvious  short-cut.  It  preserves 
the  notable  advantages  of  logistic  in  general. — the  brevity  and  precision 
of  ideographic  symbols,  and  the  consequent  assurance  of  correctness.  And 
since  it  differs  from  the  non-logistic  treatment  in  little  save  the  introduction 
of  the  logical  symbols,  it  makes  possible  the  presentation  of  the  subject  in 
hand  in  the  briefest  possible  form.  When  successful,  it  achieves  the  acme 
of  succinctness  and  clearness.  Its  shortcoming  lies  in  the  fact  that,  having 
attempted  little  which  cannot  be  accomplished  without  logistic,  it  achieves 
little  more  than  is  attained  by  the  ordinary  abstract  and  deductive  presenta- 
tion. For  what  it  is,  it  cannot  be  improved  upon;  but  those  who  are  inter- 
ested in  the  comparison  of  types  of  order,  or  the  precise  analysis  of  mathe- 
matical concepts,  will  ask  for  something  further. 

No  one  knows  how  far  the  third  method — that  of  Kempe  and  Royce — 
can  be  carried,  or  whether  the  system  S,  or  some  other  very  inclusive  type 
of  order,  will  be  found  to  contain  any  large  number,  or  all,  of  the  various 
special  orders  in  which  we  are  interested.  But  we  can  see  that,  so  far  as 
it  works,  this  method  gives  a  maximum  of  useful  result  with  a  minimum  of 
complication.  It  avoids  the  complexities  of  the  completely  analytic  method, 
yet  it  is  certain  to  disclose  whatever  analogies  exist  between  various  systems, 
by  the  fact  that  its  terms  are  allowed  to  denote  ambiguously  anything  which 
has  the  relations  in  question,  or  relations  of  precisely  that  type.  In  another 
important  respect,  also,  advantage  seems  to  lie  with  this  method.  One 
would  hardly  care  to  invent  a  new  geometry  by  the  analytical  procedure ; 
it  is  difficult  enough  to  present  one  whose  properties  are  already  familiar. 
Nor  would  one  be  likely  to  discover  the  possibility  of  a  new  system  by 
the  simple  logistic  procedure.  With  either  of  these  two  methods,  we  need 
25 


370  A  Survey  of  Symbolic  Logic 

to  know  where  we  are  going,  or  we  shall  go  nowhere.  By  contrast,  the 
third  method  is  that  of  the  pathfinder.  The  prospect  of  the  novel  is  here 
much  greater.  The  system  2  may,  probably  does,  contain  new  continents 
of  order  whose  existence  we  do  not  even  suspect.  And  some  chance  trans- 
formation may  put  us,  suddenly  and  unexpectedly,  in  possession  of  such 
previously  unexplored  fields.  The  outstanding  difficulty  of  the  method, 
apart  from  our  real  ignorance  of  its  possibilities,  seems  to  be  that  it  must 
rely  upon  devices  which  are  not  at  all  obvious.  It  may  not  tax  severely 
the  analytical  powers,  but  it  is  certain  to  tax  the  ingenuity.  Having  set 
up,  for  example,  the  general  order  of  geometrical  points,  one  may  be  at  a 
loss  how  to  specify  "lines"  having  the  properties  of  Euclidean  parallels. 
In  this  respect,  the  analytic  method  is  superior.  But  the  prospect  of 
generality  without  complexity,  wrhich  the  third  method  seems  to  offer,  is 
most  enticing. 

We  have  spoken  of  symbolic  logic,  logistic  and  mathematics.  It  may 
well  be  questioned  whether  the  method  of  logistic  does  not  admit  of  useful 
application  beyond  the  field  of  mathematics.  Symbolic  logic  is  an  instru- 
ment as  much  more  flexible  and  more  powerful  than  Aristotelian  logic  as 
modern  science  is  more  complex  than  its  medieval  counterpart.  Some  of 
the  advantages  which  might  have  accrued  to  alchemy,  had  the  alchemists 
reduced  their  speculations  to  syllogisms,  might  well  accrue  to  modern  sci- 
ence through  the  use  of  symbolic  logic.  The  use  of  ideographic  symbolism 
is  capable  of  making  quite  the  same  difference  in  the  case  of  propositions  and 
reasoning  that  it  has  already  made  in  the  case  of  numbers  and  reckoning. 
It  is  reported  that  the  early  Australian  settlers  could  buy  sheep  from  the 
Bushmen  only  by  holding  up  against  one  sheep  the  coins  or  trinkets  repre- 
senting the  price,  then  driving  off  that  sheep  and  repeating  the  process. 
It  might  be  reported  of  the  generality  of  our  thinking  that  it  is  possible  to 
get  desired  conclusions  only  by  holding  up  one  or  two  propositions,  driving 
off  the  immediate  consequences,  and  then  repeating  the  process.  Symbolic 
logic  is  capable  of  working  the  same  transformation  in  the  latter  case  that 
arithmetic  does  in  the  former.  Those  unfamiliar  with  logistic  may  not 
credit  this — but  upon  this  point  we  hesitate  to  press  the  analogy.  Certain 
it  is,  that  for  the  full  benefit  that  symbolic  logic  is  capable  of  giving,  we 
should  need  to  be  brought  up  in  it,  as  we  are  in  the  simpler  processes  of 
arithmetic.  What  the  future  may  bring  in  the  widespread  use  of  this 
new  instrument,  one  hardly  ventures  to  prophesy. 

Some  of  the  advantages  which  would  he  derived  from  the  wider  use  of 


Symbolic  Logic,  Logistic,  and  Mathematical  Method  371 

logistic  in  science,  one  can  make  out.  The  logistic  method  is  applicable 
wherever  a  body  of  fact  or  of  theory  approaches  that  completeness  and 
systematic  character  which  belongs  to  mathematical  systems.  And  by  the 
use  of  it,  the  same  assurance  of  correctness  which  belongs  to  the  mathe- 
matical portions  of  scientific  subjects  may  be  secured  for  those  portions 
which  are  not  stateable  in  terms  of  ordinary  mathematics. 

Dare  we  make  one  further  suggestion  of  the  possible  use  of  logistic  in 
science?  Since  it  seems  to  us  important,  we  shall  venture  it,  with  all  due 
apologies  for  our  ignorance  and  our  presumption.  A  considerable  part 
seems  to  be  played  in  scientific  investigation  by  imagery  which  is  more  or 
less  certainly  extraneous  to  the  real  body  of  scientific  law.  The  scientist  is 
satisfied  to  accept  a  certain  body  of  facts — directly  or  indirectly  observed 
phenomena,  "laws,"  and  hypotheses  which,  for  the  time  being  at  least, 
need  not  be  questioned.  But  beyond  this,  he  finds  a  use  for  what  is  neither 
directly  nor  indirectly  observed,  but  serves  somehow  to  represent  the  situ- 
ation. A  physicist,  for  example,  will  indulge  in  mechanical  models  of  the 
ether,  or  mechanical  models  of  the  atom  which,  however  much  he  may  hope 
to  verify  them,  he  knows  to  run  beyond  established  fact.  The  value  of 
such  imagery  is,  in  part  at  least,  its  concreteness.  The  established  relations, 
simply  in  terms  of  mathematics  and  logic,  do  not  come  to  possess  their  full 
significance  unless  they  are  vested  in  something  more  palpable.  A  great 
deal  of  what  passes  for  "hypothesis"  and  "theory"  seems  to  have,  in  part 
at  least,  this  character  and  this  value;  if  it  were  not  for  the  greater  "sug- 
gestiveness"  of  the  concrete,  much  of  this  would  have  no  reason  for  being. 
Now  whoever  has  worked  with  the  precise  and  terse  formulations  of  logistic 
realizes  that  it  is  capable  of  performing  some  of  the  offices  of  concrete 
imagery.  Its  brevity  enables  more  facts  to  be  "seen"  at  once,  thought  of 
together,  treated  as  a  single  thing.  And  a  logistic  formulation  can  be  free 
from  the  unwarranted  suggestions  to  which  other  imagery  is  liable.  Perhaps 
a  wider  use  of  logistic  would  help  to  free  science  from  a  considerable  body  of 
"hypotheses"  whose  value  lies  not  in  their  logical  implications  but  in  their 
psychological  "suggestiveness".  But  the  reader  will  take  this  conjecture 
only  for  what  it  is  worth.  What  seems  certain  is  that  for  the  presentation 
of  a  systematic  body  of  theory,  for  the  comparison  of  alternative  hypotheses 
and  theories,  and  for  testing  the  applicability  of  theory  to  observed  facts, 
logistic  is  an  instrument  of  such  power  as  to  make  its  eventual  use  almost 
certain. 

Merely  from  the  point  of  view  of  method,  the  application  of  logistic  to 


372  A  Survey  of  Symbolic  Logic 

subjects  outside  the  field  of  mathematics  needs  no  separate  discussion. 
For  when  mathematics  is  no  longer  viewed  as  the  science  of  number  and 
quantity,  but  as  it  is  viewed  by  Mr.  Russell  or  by  anyone  who  accepts  the 
alternative  definition  offered  in  this  chapter,  then  the  logistic  treatment 
of  any  subject  becomes  mathematics.  Mathematics  itself  ceases  to  have 
any  peculiar  subject  matter,  and  becomes  simply  a  method.  Logistic  is 
the  universal  method  for  presenting  exact  science  in  ideographic  symbols. 
It  is  the  "universal  mathematics"  of  Leibniz. 


Finis 


APPENDIX 
TWO  FRAGMENTS  FROM  LEIBNIZ 

(Translated  from  the  Latin  of  Gehrhardt's  text,  Die  Philosophischen  Schiiften  von  G.  W. 
Leibniz,  Band  VII,  "Scientia  Generalis.  Characteristica,"  XIX  and  XX.) 

These  two  fragments  represent  the  final  form  of  Leibniz's  "universal  calculus":  their 
date  is  not  definitely  known,  but  almost  certainly  they  were  written  after  1685.  Of  the 
two,  XX  is  in  all  respects  superior,  as  the  reader  will  see,  but  XIX  also  is  included  because 
it  contains  the  operation  of  "subtraction"  which  is  dropped  in  XX.  Leibniz's  compre- 
hension of  the  fact  that  +  and  —  (or,  in  the  more  usual  notation,  "multiplication"  and 
"division")  are  not  simple  inverses  in  this  calculus,  and  his  appreciation  of  the  complexity 
thus  introduced,  is  the  chief  point  of  interest  in  XIX.  The  distinction  of  "subtraction" 
(in  intension)  and  negation,  is  also  worthy  of  note.  It  will  be  observed  that,  in  both 
these  fragments,  A  +  B  (or  A  ®  B)  may  be  interpreted  in  two  ways:  (1)  As  "both  A 
and  B"  in  intension;  (2)  as  "either  A  or  B",  the  class  made  up  of  the  two  classes  A  and  B, 
in  extension.  The  "logical"  illustrations  mostly  follow  the  first  interpretation,  but  in  XX 
(see  esp.  scholium  to  defs.  3,  4,  5,  and  6)  there  are  examples  of  the  application  to  logical 
classes  in  extension.  The  illustration  of  the  propositions  by  the  relations  of  line-segments 
also  exhibits  the  application  to  relations  of  extension.  Attention  is  specifically  called  to 
the  parallelism  between  relations  of  intension  and  relations  of  extension  in  the  remark 
appended  to  prop.  15,  in  XX.  The  scholium  to  axioms  1  and  2,  in  XX,  is  of  particular  in- 
terest as  an  illustration  of  the  way  in  which  Leibniz  anticipates  later  logistic  developments. 

The  Latin  of  the  text  is  rather  careless,  and  constructions  are  sometimes  obscure. 
Gehrhardt  notes  (p.  232)  that  the  manuscript  contains  numerous  interlineations  and  is 
difficult  to  read  in  many  places. 

XIX 

NON   INELEGANS   SPECIMEN   DEMONSTKANDI   IN   ABSTRACTIS1 

Def.  1.  Two  terms  are  the  same  (eadem)  if  one  can  be  substituted  for  the  other  with- 
out altering  the  truth  of  any  statement  (salva  veritate).  If  we  have  A  and  B,  and  A  enters 
into  some  true  proposition,  and  the  substitution  of  B  for  A  wherever  it  appears,  results  in  a 
new  proposition  which  is  likewise  true,  and  if  this  can  be  done  for  every  such  proposition, 
then  A  and  B  are  said  to  be  the  same;  and  conversely,  if  A  and  B  are  the  same,  they  can 
be  substituted  for  one  another  as  I  have  said.  Terms  which  are  the  same  are  also  called 
coincident  (coincidentia) ;  A  and  A  are,  of  course,  said  to  be  the  same,  but  if  A  and  B  are 
the  same,  they  are  called  coincident. 

Def.  2.  Terms  which  are  not  the  same,  that  is,  terms  which  cannot  always  be  sub- 
stituted for  one  another,  are  different  (div^rsa).  Corollary.  Whence  also,  whatever  terms 
are  not  different  are  the  same. 

Charact.  I.2    A  =  B  signifies  that  A  and  B  are  the  same,  or  coincident. 

Charact.  2.3    A  4=  B,  or  B  4=  A,  signifies  that  A  and  B  are  different. 

Def.  3.  If  a  plurality  of  terms  taken  together  coincide  with  one,  then  any  one  of  the 
plurality  is  said  to  be  in  (inesse)  or  to  be  contained  in  (contineri)  that  one  with  which  they 

1  This  title  appears  in  the  manuscript,  but  Leibniz  has  afterward  crossed  it  out.     Al- 
though pretentious,  it  expresses  admirably  the  intention  of  the  fragment,  as  well  as  of  the 
next. 

2  We  write  A  =  B  where  the  text  has  A  <*>  B. 

3  We  write  A  4=  B  where  the  text  has  A  non  °o  B. 

373 


374  A  Survey  of  Symbolic  Logic 

coincide,  and  that  one  is  called  the  container.  And  conversely,  if  any  term  be  contained  in 
another,  then  it  will  be  one  of  a  plurality  which  taken  together  coincide  with  that  other. 
For  example,  if  A  and  B  taken  together  coincide  with  L,  then  A,  or  B,  will  be  called  the 
inexislent  (inexistens)  or  the  contained;  and  L  will  be  called  the  container.  However,  it 
can  happen  that  the  container  and  the  contained  coincide,  as  for  example,  if  (A  and  B)  =  L, 
and  A  and  L  coincide,  for  in  that  case  B  will  contain  nothing  which  is  different  from  A.  .  .  .* 

Scholium.  Not  every  inexistent  thing  is  a  part,  nor  is  every  container  a  whole — e.  g., 
an  inscribed  square  and  a  diameter  are  both  in  a  circle,  and  the  square,  to  be  sure,  is  a  certain 
part  of  the  circle,  but  the  diameter  is  not  a  part  of  it.  We  must,  then,  add  something  for 
the  accurate  explanation  of  the  concept  of  whole  and  part,  but  this  is  not  the  place  for  it. 
And  not  only  can  those  things  which  are  not  parts  be  contained  in,  but  also  they  can  be 
subtracted  (or  "abstracted",  detrahi);  e.  g.,  the  center  can  be  subtracted  from  a  circle 
so  that  all  points  except  the  center  shall  be  in  the  remainder;  for  this  remainder  is  the  locus 
of  all  points  within  the  circle  whose  distance  from  the  circumference  is  less  than  the  radius, 
and  the  difference  of  this  locus  from  the  circle  is  a  point,  namely  the  center.  Similarly  the 
locus  of  all  points  which  are  moved,  in  a  sphere  in  which  two  distinct  points  on  a  diameter 
remain  unmoved,  is  as  if  you  should  subtract  from  the  sphere  the  axis  or  diameter  passing 
through  the  two  unmoved  points. 

On  the  same  supposition  [that  A  and  B  together  coincide  with  L],  A  and  B  taken 
together  are  called  constituents  (constituentia),  and  L  is  called  that  which  is  constituted 
(constitutum). 

Charact.  3.     A  +  B  =  L  signifies  that  A  is  in  or  is  contained  in  L. 

Scholium.  Although  A  and  B  may  have  something  in  common,  so  that  the  two  taken 
together  are  greater  than  L  itself,  nevertheless  what  we  have  here  stated,  or  now  state,  will 
still  hold.  It  will  be  well  to  make  this  clear  by  an  example:  Let  L  denote  the  straight 
line  RX,  and  A  denote  a  part  of  it,  say  the  line  RS,  and  B  denote 

another  part,  say  the  line  XY.     Let  either  of  these  parts,  RS  or     R S X 

XY,  be  greater  than  half  the  whole  line,  RX;    then  certainly  it 

cannot  be  said  that  A  +  B  equals  L,  or  RS  +  XY  equals  RX.  For  inasmuch  as  YS  is  a 
common  part  of  RS  and  XY,  RS  +  XY  will  be  equal  to  RX  +  SY.  And  yet  it  can  truly 
be  said  that  the  lines  RS  and  XY  together  coincide  with  the  line  RS.S 

p  M  N 


'R         ^  'Y         NS  f's          \  v-  Def.  4.     If  some  term  M  is  in  A  and  also  in  B,  it 

is  said  to  be  common  to  them,  and  they  are  said  to  be 


communicating   (communicantia)  .6      But  if  they  have 
s'~~"  ''  nothing  in  common,  as  A  and  N  (the  lines  RS  and 

B  /'  XS,  for  example),  they  are  said  to  be  non-communi- 

"^  eating  (incommunicantia). 


Def.  5.  If  A  is  in  L  in  such  wise  that  there  is  another  term,  N,  in  which  belongs 
everything  in  L  except  what  is  in  A,  and  of  this  last  nothing  belongs  in  N,  then  A  is  said 
to  be  subtracted  (delrahi)  or  taken  away  (removeri),  and  N  is  called  the  remainder  (residuum). 

Charact.  4.  L  —  A  =  N  signifies  that  L  is  the  container  from  which  if  A  be  sub- 
tracted the  remainder  is  N. 

Def.  6.  If  some  one  term  is  supposed  to  coincide  with  a  plurality  of  terms  which 
are  added  (positis)  or  subtracted  (remolis),  then  the  plurality  of  terms  are  called  the  con- 
stituents, and  the  one  term  is  called  the  thing  constituted.7 

4  Lacuna  in  the  text,  followed  by  "significet  A,  significabit  Nihil". 

5  Italics  ours. 

6  The  text  here  has  "communicatia",  clearly  a  misprint. 

7  Leibniz's  idea  seems  to  be  that  if  A  +  N  =  L  then  L  is  "constituted"  by  A  and  N, 
and  also  if  L  —  A  =  N  then  L  and  A  "constitute"  N.     But  it  may  mean  that  if  L  —  A  =  N, 
then  A  and  N  "  constitute  "  L. 


Two  Fragments  from  Leibniz  375 

Scholium.  Thus  all  terms  which  are  in  anything  are  constituents,  but  the  reverse 
does  not  hold;  for  example,  L  —  A  =  N,  in  which  case  L  is  not  in  A. 

Def.  7.  Constitution  (that  is,  addition  or  subtraction)  is  either  tacit  or  expressed, — 
N  or  —  M  the  tacit  constitution  of  M  itself,  as  A  or  —  A  in  which  N  is.  The  expressed 
constitution  of  N  is  obvious.8 

Def.  8.  Compensation  is  the  operation  of  adding  and  subtracting  the  same  thing  in 
the  same  expression,  both  the  addition  and  the  subtraction  being  expressed  [as  A  +  M 
—  M].  Destruction  is  the  operation  of  dropping  something  on  account  of  compensation, 
so  that  it  is  no  longer  expressed,  and  for  M  —  M  putting  Nothing. 

Axiom  1.     If  a  term  be  added  to  itself,  nothing  new  is  constituted  or  A  +  A  =  A. 

Scholium.  With  numbers,  to  be  sure,  2  +  2  makes  4,  or  two  coins  added  to  two 
coins  make  four  coins,  but  in  that  case  the  two  added  are  not  identical  with  the  former  two ; 
if  they  were,  nothing  new  would  arise,  and  it  would  be  as  if  we  should  attempt  in  jest  to 
make  six  eggs  out  of  three  by  first  counting  3  eggs,  then  taking  away  one  and  counting 
the  remaining  2,  and  then  taking  away  one  more  and  counting  the  remaining  1. 

Axiom  2.  If  the  same  thing  be  added  and  subtracted,  then  however  it  enter  into  the 
constitution  of  another  term,  the  result  coincides  with  Nothing.  Or  A  (however  many 
times  it  is  added  in  constituting  any  expression)  —  A  (however  many  times  it  is  subtracted 
from  that  same  expression)  =  Nothing. 

Scholium.  Hence  A  —  A  or  (A  +  A  — )  —  A  or  A  (A  +  A),  ete.  =  Nothing.  For 
by  axiom  1,  the  expression  in  each  case  reduces  to  A  —A. 

Postulate  1.  Any  plurality  of  terms  whatever  can  be  added  to  constitute  a  single 
term;  as  for  example,  if  we  have  A  and  B,  we  can  write  A  +  B,  and  call  this  L. 

Post.  2.  Any  term,  A,  can  be  subtracted  from  that  in  which  it  is,  namely  A  +  B 
or  L,  if  the  remainder  be  given  as  B,  which  added  to  A  constitutes  the  container  L — that 
is,  on  this  supposition  [that  A  +  B  =  L]  the  remainder  L  —  A  can  be  found. 

Scholium.  In  accordance  with  this  postulate,  we  shall  give,  later  on,  a  method  for 
finding  the  difference  between  two  terms,  one  of  which,  A,  is  contained  in  the  other, 'L, 
even  though  the  remainder,  which  together  with  A  constitutes  L,  should  not  be  given — 
that  is,  a  method  for  finding  L  —  A,  or  A  +  B  —  A,  although  A  and  L  only  are  given, 
and  B  is  not. 

THEOREM  1 

Terms  which  are  the  same  with  a  third,  are  the  same  with  each  other. 

If  A  =  B  and  B  =  C,  then  A  =  C.  For  if  in  the  proposition  A  =  B  (true  by  hyp.) 
C  be  substituted  for  B  (which  can  be  done  by  def.  1,  since,  by  hyp.,  B  =  C),  the  result 
is  A  =  C.  Q.E.D. 

THEOREM  2 

//  one  of  two  terms  which  are  the  same  be  different  from  a  third  term,  then  the  other  of  the 
two  will  be  different  from  it  also. 

If  A  =  B  and  B  =J=  C,  then  A  4=  C.  For  if  in  the  proposition  B  =f=  C  (true  by  hyp.) 
A  be  substituted  for  B  (which  can  be  done  by  def.  1,  since,  by  hyp.,  A  =  B),  the  result  is 
A  *  C.  Q.E.D. 

[Theorem  in  the  margin  of  the  manuscript.] 

Here  might  be  inserted  the  following  theorem:  Whatever  is  in  one  of  two  coincident 
terms,  is  in  the  other  also. 

If  A  is  in  B  and  B  =  C,  then  also  A  is  in  C.  For  in  the  proposition  A  is  in  B  (true 
by  hyp.)  let  C  be  substituted  for  B. 

THEOREM  3 

//  terms  which  coincide  be  added  to  the  same  term,  the  results  will  coincide. 

If  A  =  B,  then  A  +  C  =  B  +  C.    For  if  in  the  proposition  A  +  C  =  A  +  C  (true 

8  This  translation  is  literal:  the  meaning  is  obscure,  but  see  the  diagram  above. 


376  A  Survey  of  Symbolic  Logic 

per  se)  you  substitute  B  for  A  in  one  place  (which  can  be  done  by  def.  1,  since  A  =  B), 
it  gives  A  +  C  =  B  +  C.    Q.E.D. 

COROLLARY.  //  terms  which  coincide  be  added  to  terms  which  coincide,  the  results  will 
coincide.  If  A  =  B  and  L  =  M,  then  A  -f  L  =  B  +  M .  For  (by  the  present  theorem) 
since  L  =  M,  A  +  L  —  A  +  M,  and  in  this  assertion  putting  B  for  A  in  one  place  (since 
by  hyp.  A  =  B)  gives  A  +  L  =  B  +  M.  Q.E.D. 

THEOREM  4 

A  container  of  the  container  is  a  container  of  the  contained',  or  if  that  in  which  some- 
thing is,  be  itself  in  a  third  thing,  then  that  which  is  in  it  will  be  in  that  same  third  thing — 
that  is,  if  A  is  in  B  and  B  is  in  C,  then  also  A  is  in  C. 

For  A  is  in  B  (by  hyp.),  hence  (by  def.  3  or  charact.  3)  there  is  some  term,  which  we 
may  call  L,  such  that  A  +  L  =  B.  Similarly,  since  B  is  in  C  (by  hyp.),  B  +  M  =  C, 
and  in  this  assertion  putting  A  +  L  f or  B  (since  we  show  that  these  coincide)  we  have 
A  +  L  +  M  =  C.  But  putting  N  f  or  L  +  M  (by  post.  1)  we  have  A  +  N  =  C.  Hence 
(by  def.  3)  A  is  in  C.  Q.E.D. 

THEOREM  5 

Whatever  contains  terms  individually  contains  also  that  which  is  constituted  of  them. 

If  A  is  in  C  and  B  is  in  C,  then  A  +  B  (constituted  of  A  and  B,  def.  4)  is  in  C.  For 
since  A  is  in  C,  there  will  be  some  term  M  such  that  A  +  M  =  C  (by  def.  3).  Similarly, 
since  B  is  in  C,  B  +  N  =  C.  Putting  these  together  (by  the  corollary  to  th.  3) ,  we  have 
A  +  M  +  B  +  N  =  C  +  C.  But  C  +  C  =  C  (by  ax.  1),  hence  A+M+B  +  N  =  C. 
And  therefore  (by  def.  3)  A  +  B  is  in  C.  Q.E.D.9 

THEOREM  6 

Whatever  is  constituted  of  terms  which  are  contained,  is  in  that  which  is  constituted  of  the 
containers. 

If  A  is  in  M  and  B  is  in  N,  then  A  +  B  is  in  M  +  N.  For  A  is  in  M  (by  hyp.)  and  M 
is  in  M  +  N  (by  def.  3),  hence  A  is  in  M  +  N  (by  th.  4).  Similarly,  B  is  in  N  (by  hyp.) 
and  N  is  in  M  +  N  (by  def.  3),  hence  B  is  in  M  +  N  (by  th.  4).  But  if  A  is  in  M  +  N 
and  B  is  in  M  +  N,  then  also  (by  th.  5)  A  +  B  is  in  M  +  N.  Q.E.D. 

THEOREM  7 

//  any  term  be  added  to  that  in  which  it  is,  then  nothing  new  is  constituted;  or  if  B  is  in  A, 
then  A  +  B  =  A. 

For  if  B  is  in  A,  then  [for  some  C]  B  +  C  =  A  (def.  3).  Hence  (by  th.  3)  A  +  B 
=  B  +  C  +  B  =  B  +  C  (by  ax.  1)  =  A  (by  the  above).  Q.E.D. 

CONVERSE  OF  THE  PRECEDING  THEOREM 

//  by  the  addition  of  any  term  to  another  nothing  new  is  constituted,  then  the  term  added 
is  in  the  other. 

If  A  +  B  =  A,  then  B  is  in  A;  for  B  is  in  A  +  B  (def.  3),  and  A  +  B  =  A  (by  hyp.). 
Hence  B  is  in  A  (by  the  principle  which  is  inserted  between  ths.  2  and  3).  Q.E.D. 

THEOREM  8 

//  terms  which  coincide  be  subtracted  from  terms  which  coincide,  the  remainders  will 
coincide. 

If  A  =  L  and  B  =  M ,  then  A  —  B  =  L  -  M.     For  A  -  B  =  A  —  B  (true  per  se), 

9  In  the  margin  of  the  manuscript  at  this  point  Leibniz  has  an  untranslatable  note, 
the  sense  of  which  is  to  remind  him  that  he  must  insert  illustrations  of  these  propositions  in 
common  language. 


Two  Fragments  from  Leibniz  377 

and  the  substitution,  on  one  or  the  other  side,  of  L  for  A  and  M  for  B,  gives  A  —  B  =  L 
-  M.    Q.E.D. 

[Note  in  the  margin  of  the  manuscript.]  In  dealing  with  concepts,  subtraction  (de- 
tractio)  is  one  thing,  negation  another.  For  example,  "non-rational  man"  is  absurd  or 
impossible.  But  we  may  say;  An  ape  is  a  man  except  that  it  is  not  rational.  [They 
are]  men  except  in  those  respects  in  which  man  differs  from  the  beasts,  as  in  the  case  of 
Grotius's  Jumbo10  (Homines  nisi  qua  bestiis  differt  homo,  ut  in  Jambo  Grotii).  "Man" 

—  "rational"  is  something  different  from  "non-rational  man".     For  "man"  —  "rational" 
=  "brute".     But  "non-rational  man"  is  impossible.     "Man"  —  "animal"  —  "rational" 
is  Nothing.     Thus  subtractions  can  give  Nothing  or  simple  non-existence — even  less  than 
nothing — but  negations  can  give  the  impossible.11 

THEOREM  9 

(1)  From  an  expressed  compensation,  the  destruction  of  the  term  compensated  follows, 
provided  nothing  be  destroyed  in  the  compensation  which,  being  tacitly  repeated,  enters 
into  a  constitution  outside  the  compensation  [that  is,  +  N  —  N  appearing  in  an  expression 
may  be  dropped,  unless  N  be  tacitly  involved  in  some  other  term  of  the  expression]; 

(2)  The  same  holds  true  if  whatever  is  thus  repeated  occur  both  in  what  is  added  and 
in  what  is  subtracted  outside  the  compensation; 

(3)  If  neither  of  these  two  obtain,  then  the  substitution  of  destruction  for  compensa- 
tion [that  is,  the  dropping  of  the  expression  of  the  form  -f-  N  —  N]  is  impossible. 

Case  1.  If  A  +  N  —  M  —  N  =  A  —  M,  and  A,  N,  and  M  be  non-communicating. 
For  here  there  is  nothing  in  the  compensation  to  be  destroyed,  +  N  —  N,  which  is  also 
outside  it  in  A  or  M — that  is,  whatever  is  added  in  +  N.,  however  many  times  it  is  added, 
is  in  +  N,  and  whatever  is  subtracted  in  —  N,  however  many  times  it  is  subtracted,  is  in 

—  N.    Therefore  (by  ax.  2)  for  +  N  —  N  we  can  put  Nothing. 

Case  2.  If  A  +  B  —  B  —  G  =  F,  and  whatever  is  common  both  to  A  +  B  [i.  e., 
to  A  and  B]  and  to  G  and  B,  is  M,  then  F  =  A  —  G.  In  the  first  place,  let  us  suppose 
that  whatever  A  and  G  have  in  common,  if  they  have  anything  in  common,  is  E,  so  that 
if  they  have  nothing  in  common,  then  E  =  Nothing.  Thus  [to  exhibit  the  hypothesis 
of  the  case  more  fully]  A  =  E  +  Q  +  M,  B  =  N  +  M,  and  G  =  E  +  H  +  M,  so  that 
F=E  +  Q  +  M  +  N-N-M-H-M,  where  all  the  terms  E,  Q,  M,  N,  and  H  are 
non-communicating.  Hence  (by  the  preceding  case)  F  =  Q  —  H=E+Q+M—  E 
-H  -  M  =  A  -G. 

Case  3.  If  A  -f-  B  —  B  —  D  =  C,  and  that  which  is  common  to  A  and  B  does  not 
coincide  with  that  which  is  common  to  B  +  D  [i.  e.,  to  B  and  D],  then  we  shall  not  have 
C  =  A  -  D.  FoT\etB=E  +  F  +  G,  and  A  =  H  +  E,  and  D  =  K  +  F,  so  that  these 
constituents  are  no  longer  communicating  and  there  is  no  need  for  further  resolution. 
Then  C  =  H+E  +  F  +  G~E-F-G-K-F,  that  is  (by  case  1)  C  =  H  -  K, 
which  is  not  =  A  —  D  (since  A  —  D  =  H  +  E  —  K  —  F),  unless  we  suppose,  contrary 
to  hypothesis,  that  E  =  F — that  is  that  B  and  A  have  something  in  common  which  is  also 
common  to  B  and  D.  This  same  demonstration  would  hold  even  if  A  and  D  had 
something  in  common. 

10  Apparently  an  allusion  to  some  description  of  an  ape  by  Grotius. 

11  This  is  not  an  unnecessary  and  hair-splitting  distinction,  but  on  the  contrary,  per- 
haps the  best  evidence  of  Leibniz's  accurate  comprehension  of  the  logical  calculus  which 
appears  in  the  manuscripts.     It  has  been  generally  misjudged  by  the  commentators,  because 
the  commentators  have  not  understood  the  logic  of  intension.     The  distinction  of  the 
merely  non-existent  and  the  impossible  (self-contradictory  or  absurd)  is  absolutely  essential 
to  any  calculus  of  relations  in  intension.     And  this  distinction  of  subtraction  (or  in  the  more 
usual  notation,  division)  from  negation,  is  equally  necessary.     It  is  by  the  confusion  of 
these  two  that  the  calculuses  of  Lambert  and  Castillon  break  down. 


378  A  Survey  of  Symbolic  Logic 

THEOREM  10 

A  subtracted  term  and  the  remainder  are  non-communicating. 

If  L  —  A  =  N,  1  affirm  that  A  and  2V  have  nothing  in  common.  For  by  the  definition 
of  "subtraction"  and  of  "remainder",  everything  in  L  remains  in  N  except  that  which  is 
in  A,  and  of  this  last  nothing  remains  in  N. 

THEOREM  11 

Of  that  which  is  in  two  communicating  terms,  whatever  part  is  common  to  both  and  the 
two  exclusive  parts  are  three  non-communicating  terms. 

If  A  and  B  be  communicating  terms,  and  A  =  P  +  M  and  B  =  N  +  M ,  so  that 
whatever  is  in  A  and  B  both  is  in  M,  and  nothing  of  that  is  in  P  or  N,  then  P,  M,  and  N 
are  non-communicating.  For  P,  as  well  as  N,  is  non-communicating  with  M,  since  what- 
ever is  in  M  is  in  A  and  B  both,  and  nothing  of  this  description  is  in  P  or  N.  Then  P  and 
N  are  non-communicating,  otherwise  what  is  common  to  them  would  penetrate  into  A 
and  B  both. 

PROBLEM 

To  add  non-coincident  terms  to  given  coincident  terms  so  that  the  resulting  terms  shall 
coincide. 

If  A  =  A,  I  affirm  that  it  is  possible  to  find  two  terms,  B  and  N,  such  that  B  =J=  N 
and  yet  A  +  B  =  A  +  N. 

Solution.  Choose  some  term  M  which  shall  be  contained  in  A  and  such  that,  N 
being  chosen  arbitrarily,  M  is  not  contained  in  N  nor  N  in  M,  and  let  B  =  M  +  N.  And 
this  will  satisfy  the  requirements.  Because  B  =  M  +  N  (by  hyp.)  and  M  and  N  are 
neither  of  them  contained  in  the  other  (by  hyp.),  and  yet  A  +  B  =  A  +  N,  since  A  +  B 
=  A  +  M  +  N  and  (by  th.  7,  since,  by  hyp.,  M  is  in  A)  this  is  =  A  +  N. 

THEOREM  12 

Where  non-communicating  terms  only  are  involved,  whatever  terms  added  to  coincident 
terms  give  coincident  terms  will  be  themselves  coincident. 

That  is,  if  A  +  B  =  C  +  D  and  A  =  C,  then  B  =  D,  provided  that  A  and  B,  as 
well  as  C  and  D,  are  non-communicating.  For  A  +  B  —  C  —  C  -{•  D  —  C  (by  th.  8) ; 
but  A  +  B  -  C  =  A  +  B  -  A  (by  hyp.  that  A  =  C),  and  A  +  B  -  A  =  B  (by  th.  9, 
case  1,  since  A  and  B  are  non-communicating),  and  (for  the  same  reason)  C  +  D  —  C  =  D. 
Hence  B  =  D.  Q.E.D. 

THEOREM  13 

In  general;  if  other  terms  added  to  coincident  terms  give  coincident  terms,  then  the  terms 
added  are  communicating. 

If  A  and  A  coincide  or  are  the  same,  and  A  +  B  =  A  +  N,  I  affirm  that  B  and  N 
are  communicating.  For  if  A  and  B  are  non-communicating,  and  A  and  N  also,  then 
B  —  N  (by  the  preceding  theorem).  Hence  B  and  N  are  communicating.  But  if  A  and  B 
are  communicating,  let  A  —  P  +  M  and  B  =  Q  +  M,  putting  M  for  that  which  is  common 
to  A  and  B  and  nothing  of  this  description  in  P  or  Q.  Then  (by  ax.  1)  A  +  B  =  P  +  Q 
+  M  =  P  +  M  +  N.  But  P,  Q,  and  M  are  non-communicating  (by  th.  11).  Therefore, 
if  A?"  is  non-communicating  with  A — that  is,  with  P  +  M — then  (by  the  preceding  theorem) 
it  results  from  P  +  Q+M  =  P  +  M  +  N  that  Q  =  N.  Hence  N  is  in  B;  hence  N  and 
B  are  communicating.  But  if,  on  the  same  assumption  (namely,  that  P  +  Q  +  M 
=  P  +  M  +  N,  or  A  is  communicating  with  B)  N  also  be  communicating  with  P  +  M 
or  A,  then  either  N  will  be  communicating  with  M,  from  which  it  follows  that  it  will  be 
communicating  with  B  (which  contains  M)  and  the  theorem  will  hold,  or,  N  will  be  com- 
municating with  P,  and  in  that  case  we  shall  in  similar  fashion  let  P  =  G  +  H  and  N  =  F 
+  H,  so  that  G,  F,  and  H  are  non-communicating  (according  to  th.  11),  and  from  P  +  Q 


Two  Fragments  from  Leibniz  379 


+  M  =  P  +  M  +  N  we  get  G  +  H+Q+M  =  G  +  H  +  M  +  F  +  H.  Hence  (by 
the  preceding  theorem)  Q  =  F.  Hence  N  (  =  F  +  H)  and  B  (  =  Q  +  M)  have  something 
in  common.  Q.E.D. 

Corollary.  From  this  demonstration  we  learn  the  following:  If  any  terms  be  added 
to  the  same  or  coincident  terms,  and  the  results  coincide,  and  if  the  terms  added  are  each 
non-communicating  with  that  to  which  it  is  added,  then  the  terms  added  [to  the  same  or 
coincidents]  coincide  with  each  other  (as  appears  also  from  th.  12).  But  if  one  of  the 
terms  added  be  communicating  with  that  to  which  it  is  added,  and  the  other  not,  then  [of 
these  two  added  terms]  the  non-communicating  one  will  be  contained  in  the  communicating 
one.  Finally,  if  each  of  the  terms  is  communicating  with  that  to  which  it  is  added,  then  at 
least  they  will  be  communicating  with  each  other  (although  in  another  connection  it  would 
not  follow  that  terms  which  communicate  with  a  third  communicate  with  each  other). 
To  put  it  in  symbols:  A  +  B  =  A  +  N.  If  A  and  B  are  non-communicating,  and  A 
and  N  likewise,  then  B  =  N.  HA  and  B  are  communicating  but  A  and  N  are  non-com- 
municating, then  A"  is  in  B.  And  finally,  if  B  communicates  with  A,  and  likewise  N  com- 
municates with  A,  then  B  and  2V  at  least  communicate  with  each  other. 

XX 

Def.  1.  Terms  which  can  be  substituted  for  one  another  wherever  we  please  without 
altering  the  truth  of  any  statement  (salva  veritate),  are  the  same  (eadem)  or  coincident 
(coincidentia).  For  example,  "triangle"  and  "trilateral",  for  in  every  proposition  demon- 
strated by  Euclid  concerning  "triangle",  "trilateral"  can  be  substituted  without  loss  of 
truth. 


A  =  B12  signifies  that  A  and  B  are  the 
same,  or  as  we  say  of  the  straight  line  XY 
and  the  straight  line  YX,  XY  =  YX,  or  the 
shortest  path  of  a  [point]  moving  from  X  to 
Y  coincides  with  that  from  Y  to  X. 


Def.  2.  Terms  which  are  not  the  same,  that  is,  terms  which  cannot  always  be  sub- 
stituted for  one  another,  are  different  (diver  sa).  Such  are  "circle"  and  "triangle",  or 
"square"  (supposed  perfect,  as  it  always  is  in  Geometry)  and  "equilateral  quadrangle", 
for  we  can  predicate  this  last  of  a  rhombus,  of  which  "square"  cannot  be  predicated. 

A  =j=  B13  signifies  that  A  and  B  are  different,  as  for  example,  R  __  Y  _  *?  _  ^ 
the  straight  lines  XY  and  RS. 

Prop.  1.  If  A  =  B,  then  also  B  =  A.  If  anything  be  the  same  with  another,  then 
that  other  will  be  the  same  with  it.  For  since  A  =  B  (by  hyp.),  it  follows  (by  def.  1)  that 
in  the  statement  A  =  B  (true  by  hyp.)  B  can  be  substituted  for  A  and  A  for  B;  hence  we 
have  B  =  A. 

Prop.  2.  If  A  =f=  B,  then  also  B  ^  A.  If  any  term  be  different  from  another,  then  that 
other  will  be  different  from  it.  Otherwise  we  should  have  B  —  A,  and  in  consequence  (by 
the  preceding  prop.)  A  =  B,  which  is  contrary  to  hypothesis. 

Prop.  3.  If  A  =  B  and  B  =  C,  then  A  =  C.  Terms  which  coincide  with  a  third  term 
coincide  with  each  other.  For  if  in  the  statement  A  =  B  (true  by  hyp.)  C  be  substituted 
for  B  (by  def.  1,  since  A  =  B),  the  resulting  proposition  will  be  true. 

Coroll.  HA=B  and  B  =  C  and  C  =  D,  then  A  =  D;  and  so  on.  For  A  =  B  =  C, 
hence  A  =  C  (by  the  above  prop.).  Again,  A  =  C  =  D;  hence  (by  the  above  prop.) 
A  =  D. 

Thus  since  equal  things  are  the  same  in  magnitude,  the  consequence  is  that  things 
equal  to  a  third  are  equal  to  each  other.  The  Euclidean  construction  of  an  equilateral 
triangle  makes  each  side  equal  to  the  base,  whence  it  results  that  they  are  equal  to  each 

12  A  =  B  f  or  A  oo  B,  as  before. 

13  A  4=  B  for  A  non  oo  B,  as  before. 


380  A  Survey  of  Symbolic  Logic 

other.  If  anything  be  moved  in  a  circle,  it  is  sufficient  to  show  that  the  paths  of  any  two 
successive  periods,  or  returns  to  the  same  point,  coincide,  from  which  it  is  concluded  that 
the  paths  of  any  two  periods  whatever  coincide. 

Prop.  4.  If  A  =  B  and  B  =J=  C,  then  A  ^  C.  If  of  two  things  which  are  the  same  with 
each  other,  one  differ  from  a  third,  then  the  other  also  will  differ  from  that  third.  For  if  in  the 
proposition  B  4=  C  (true  by  hyp.)  A  be  substituted  for  B,  we  have  (by  def.  1,  since  A  =  B) 
the  true  proposition  A  4=  C. 

Def.  3.  A  is  in  L,  or  L  contains  A,  is  the  same  as  to  say  that  L  can  be  made  to  coin- 
cide with  a  plurality  of  terms,  taken  together,  of  which  A  is  one. 

Def.  4.  Moreover,  all  those  terms  such  that  whatever  is  in  them  is  in  L,  are  together 
called  components  (componentia)  with  respect  to  the  L  thus  composed  or  constituted. 

B  ©  N  =  L  signifies  that  B  is  in  L;  and  that  B  and  N  together  compose  or  constitute 
L.14  The  same  thing  holds  for  a  larger  number  of  terms. 

Def.  5.  I  call  terms  one  of  which  is  in  the  other  subalternates  (suballernantia),  as  A 
and  B  if  either  A  is  in  B  or  B  is  in  A. 

Def.  6.     Terms  neither  of  which  is  in  the  other  [I  call]  disparate  (disparata). 

Axiom  1.    B  ®  N  =  N  ®  B,  or  transposition  here  alters  nothing. 

Post.  2.  Any  plurality  of  terms,  as  A  and  B,  can  be  added  to  compose  a  single  term, 
A  ©  B  or  L. 

Axiom  2.  A  ©  A  =  A.  If  nothing  new  be  added,  then  nothing  new  results,  or 
repetition  here  alters  nothing.  (For  4  coins  and  4  other  coins  are  8  coins,  but  not  4  coins 
and  the  same  4  coins  already  counted). 

Prop.  5.  If  A  is  in  B  and  A  =  C,  then  C  is  in  B.  That  which  coincides  with  the  in- 
existent,  is  inexistent.  For  in  the  proposition,  A  is  in  B  (true  by  hyp.),  the  substitution 
of  C  for  A  (by  def.  1  of  coincident  terms,  since,  by  hyp.,  A  =  C)  gives,  C  is  in  B. 

Prop.  6.  //  C  is  in  B  and  A  =  B,  then  C  is  in  A.  Whatever  is  in  one  of  two  coincident 
terms,  is  in  the  other  also.  For  in  the  proposition,  C  is  in  B,  the  substitution  of  A  for  C 
(since  A  =  C)  gives,  A  is  in  B.  (This  is  the  converse  of  the  preceding.) 

Prop.  7.  A  is  in  A.  Any  term  whatever  is  contained  in  itself.  For  A  is  in  A  ©  A 
(by  def.  of  "inexistent",  that  is,  by  def.  3)  and  A  ©  A  =  A  (by  ax.  2).  Therefore  (by 
prop.  6),  A  is  in  A. 

Prop.  8.  If  A  =  B,  then  A  is  in  B.  Of  terms  which  coincide,  the  one  is  in  the  other. 
This  is  obvious  from  the  preceding.  For  (by  the  preceding)  A  is  in  A — that  is  (by  hyp.), 
in  B. 

Prop.  9.  If  A  =  B,  then  A  ffi  C  =  B  ©  C.  If  terms  which  coincide  be  added  to  the 
same  term,  the  results  will  coincide.  For  if  in  the  proposition,  A  ©  C  =  A  ©  C  (true 
per  se),  for  A  in  one  place  be  substituted  B  which  coincides  with  it  (by  def.  1),  we  have 
A  ©  C  =  B  ©  C. 

A®_C 

/   ,_A_^  \  A.  "triangle"   "I 

/,'''  "^>-N  C    \  /"coincide 

//  \    ,'' N\  B  "trilateral" } 

i* 

A     &    /^  t{  a/tTiiloi-iai-ol    -f  T-ionrrl^i  " 

'}' 


^^   ^'  A  ©  C"  equilateral  triangle" 

\  x^  „-''  C     ''  5  ©  C  "equilateral  trilateral"  /-coincide 

"" 


V  T>  S 

"^  /' 

~B~9~C 

14  In  this  fragment,  as  distinguished  from  XIX,  the  logical  or  "real"  sum  is  repre- 
sented by  ©.  Leibniz  has  carelessly  omitted  the  circle  in  many  places,  but  we  write  ©• 
wherever  this  relation  is  intended. 


Two  Fragments  from  Leibniz  381 

Scholium.  This  proposition  cannot  be  converted — much  less,  the  two  which  follow. 
A  method  for  finding  an  illustration  of  this  fact  will  be  exhibited  below,  in  the  problem  which 
is  prop.  23. 

Prop.  10.  //  A  =  L  and  B  =  M,  then  A  ©  B  =  L  ©  M .  If  terms  which  coincide 
be  added  to  terms  which  coincide,  the  results  will  coincide.  For  since  B  =  M,  A  ©  B  =  A  ©  M 
(by  the  preceding),  and  putting  L  for  the  second  A  (since,  by  hyp.,  A  =  L)  we  have  A  ©  B 

=  L  ©  M. 


A  "triangle",  and  L  "trilateral"  coin-  / 

cide.    B  ""regular"  coincides  with  M  "most  J/ 

capacious  of  equally-many-sided  figures  with          R{ 
equal  perimeters".     " Regular  triangle "  coin-  Vs 

cides  with  "most  capacious  of  trilateral  mak- 
ing  equal  peripheries  out  of  three  sides". 


Scholium.  This  proposition  cannot  be  converted,  for  if  A  ©  B  =  L  ©  M  and  A  =  L, 
still  it  does  not  follow  that  B  =  M,  —  and  much  less  can  the  following  be  converted. 

Prop.  11.  //  A  =  L  and  B  =  M  and  C  =  N,  then  A®B®C=L®M®N. 
And  so  on.  //  there  be  any  number  of  terms  under  consideration,  and  an  equal  number  of 
them  coincide  with  an  equal  number  of  others,  term  for  term,  then  that  which  is  composed  of  the 
former  coincides  with  that  which  is  composed  of  the  latter.  For  (by  the  preceding,  since 
A  =  L  and  B  =  M)  we  have  A  ©  B  —  L  ©  M.  Hence,  since  C  =  N,  we  have  (again 
by  the  preceding)  A@B@C  =  L®M®N. 

Prop.  12.  //  B  is  in  L,  then  A  ©  B  will  be  in  A  ©  L.  If  the  same  term  be  added  to 
what  is  contained  and  to  what  contains  it,  the  former  result  is  contained  in  the  latter.  For 
L  =  B  ©  N  (by  def.  of  "inexistent"),  and  A  ©  B  is  in  B  ©  N  ©  A  (by  the  same),  that 
is,  A  ©  B  is  in  L  ©  A. 


B  "equilateral",  L  "regular",  A  "quad- 

//'  v\  \  rilateral".     "Equilateral"  is  in  or  is  attribute 

v  \  \  of  "regular".     Hence  "equilateral  quadrilat- 

(•^       Y  _  r-  _  V^         W-'  eral"  is  in  "regular  quadrilateral"  or  "perfect 

\         \N       /          /  /  square".      YS  is  in  RX.     Hence  RT  ©  YS, 

Vx--__JSXl     ^''  /  or  RS,  is  in  RT  ©  RX,  or  in  RX. 

A      X~#          ^'' 

"~r" 

Scholium.  This  proposition  cannot  be  converted;  for  if  A  ©  B  is  in  A  ©  L,  it  does 
not  follow  that  B  is  in  L. 

Prop.  13.  //  L  ©  B  =  L,  then  B  is  in  L.  If  the  addition  of  any  term  to  another  does 
not  alter  that  other,  then  the  term  added  is  in  the  other.  For  B  is  in  L  ©  B  (by  def.  of  "in- 
existent")  and  L  ©  B  =  L  (by  hyp.),  hence  (by  prop.  6)  B  is  in  L. 


RY  ©  RX  =  RX.    Hence  RY  is  in  RX.  R  !  _  T  _  ^  „ 

RY  is  in  RX.    Hence  RY  ©  RX  =  RX.  \T  '  ! 


B~®L 


382  A  Survey  of  Symbolic  Logic 

Let  L  be  "parallelogram"  (every  side  of  which  is  parallel  to  some  side),15  B  be  "quadri- 
lateral". 

"Quadrilateral  parallelogram"  is  in  the  same  as  "parallelogram". 
Therefore  to  be  quadrilateral  is  in  [the  intension  of]  "parallelogram". 
Reversing  the  reasoning,  to  be  quadrilateral  is  in  "parallelogram". 
Therefore,  "quadrilateral  parallelogram"  is  the  same  as  "parallelogram". 

Prop.  14.  If  B  is  in  L,  then  L  ©  B  =  L.  Subalternates  compose  nothing  new;  or  if 
any  term  which  is  in  another  be  added  to  it,  it  will  produce  nothing  different  from  that  other. 
(Converse  of  the  preceding.)  If  B  is  in  L,  then  (by  def.  of  "inexistent")  L  =  B  ©  P.  Hence 
(by  prop.  9)L@B=B®P®B,  which  (by  ax.  2)  is  =  B  ©  P,  which  (by  hyp.)  is  =  L. 

Prop.  15.  If  A  is  in  B  and  B  is  in  C,  then  also  A  is  in  C.  What  is  contained  in  the 
contained,  is  contained  in  the  container.  For  A  is  in  B  (by  hyp.),  hence  A  ©  L  =  B  (by 
def.  of  "inexistent").  Similarly,  since  B  is  in  C,  B  ©  M  =  C,  and  putting  A  ©  L  for  B 
in  this  statement  (since  we  have  shown  that  these  coincide),  we  have  A  ©  L  ©  M  =  C. 
Therefore  (by  def.  of  "inexistent")  A  is  in  C. 

R  T  S          X  RT  is  in  RS,  and  RS  in  RX. 

V— "A  /        ~~7  Hence  RT  is  in  RX. 

/  A  "quadrilateral",  B  "parallelogram", 

yc  C  "rectangle". 

To  be  quadrilateral  is  in  [the  intension  of]  "parallelogram",  and  to  be  parallelogram 
is  in  "rectangle"  (that  is,  a  figure  every  angle  of  which  is  a  right  angle).  If  instead  of 
concepts  per  se,  we  consider  individual  things  comprehended  by  the  concept,  and  put  A 
for  "rectangle",  B  for  "parallelogram",  C  for  "quadrilateral",  the  relations  of  these  can 
be  inverted.  For  all  rectangles  are  comprehended  in  the  number  of  the  parallelograms, 
and  all  parallelograms  in  the  number  of  the  quadrilaterals.  Hence  also,  all  rectangles  are 
contained  amongst  (in)  the  quadrilaterals.  In  the  same  way,  all  men  are  contained  amongst 
(in)  all  the  animals,  and  all  animals  amongst  all  the  material  substances,  hence  all  men 
are  contained  amongst  the  material  substances.  And  conversely,  the  concept  of  material 
substance  is  in  the  concept  of  animal,  and  the  concept  of  animal  is  in  the  concept  of  man. 
For  to  be  man  contains  [or  implies]  being  animal. 

Scholium.     This  proposition  cannot  be  converted,  and  much  less  can  the  following. 

Coroll.    If  A  ©  N  is  in  B,  N  also  is  in  B.     For  N  is  in  A  ©  N  (by  def.  of  " inexistent "). 

Prop.  16.  //  A  is  in  B  and  B  is  in  C  and  C  is  in  D,  then  also  A  is  in  D.  And  so  on. 
That  which  is  contained  in  what  is  contained  by  the  contained,  is  in  the  container.  For  if  A 
is  in  B  and  B  is  in  C,  A  also  is  in  C  (by  the  preceding).  Whence  if  C  is  in  D,  then  also 
(again  by  the  preceding)  A  is  in  D. 

Prop.  17.  //  A  is  in  B  and  B  is  in  A,  then  A  =  B.  Terms  which  contain  each  other 
coincide.  For  if  A  is  in  B,  then  A  ©  N  =  B  (by  def.  of  "inexistent").  But  B  is  in  A 
(by  hyp.),  hence  A  ©  N  is  in  A  (by  prop.  5).  Hence  (by  coroll.  prop.  15)  N  also  is  in  A. 
Hence  (by  prop.  14)  A  =  A  ©  N,  that  is,  A  =  B. 

B_ 

RT,  N;  RS,  A;  SR  ©  RT,  B.  ,'''  ^^ 

To  be  trilateral  is  in   [the  intension  of]  //''  \ 

"triangle",  and  to  be  triangle  is  in  "trilat- 
eral". Hence  "triangle"  and  "trilateral" 
coincide.  Similarly,  to  be  omniscient  is  to  be  \  "  "  / 

omnipotent.  \x  ,'' 

15 Leibniz  uses  "parallelogram"  in  its  current  meaning,  though  his  language^may 
suggest  a  wider  use. 


Two  Fragments  from  Leibniz  383 

Prop.  18.  //  A  is  in  L  and  B  is  in  L,  then  also  A  ©  B  is  in  L.  What  is  composed  of 
two,  each  contained  in  a  third,  is  itself  contained  in  that  third.  For  since  A  is  in  L  (by  hyp.), 
it  can  be  seen  that  A  ©  M  =  L  (by  def.  of  "inexistent").  Similarly,  since  B  is  in  L, 
it  can  be  seen  that  B  ©  N  =  L.  Putting  these  together,  we  have  (by  prop.  10)  A  ©  M 
®B®N  =  L®L.  Hence  (by  ax.  2)18  A®M@B®N  =  L,  Hence  (by  def.  of 

16  The  number  of  the  axiom  is  given  in  the  text  as  5,  a  misprint, 
"inexistent")  A  ©  B  is  in  L. 


RYS  is  in  RX.  /  \ 

YSTisinRX.  Bfr £ £ f 

Hence  BT  is  in  BX.  *\  \         /  J 


A  "equiangular",  B  "equilateral",  A  ©  B  "equiangular  equilateral"  or  "regular", 
L  "square".  "Equiangular"  is  in  [the  intension  of]  "square",  and  "equilateral"  is  in 
"square".  Hence  "regular"  is  in  "square". 

Prop.  19.  //  A  is  in  L  and  B  is  in  L  and  C  is  in  L,  then  A  ©  B  ©  C  is  in  L.  And 
so  on.  Or  in  general,  whatever  contains  terms  individually,  contains  also  what  is  composed  of 
them.  For  A  ©  B  is  in  L  (by  the  preceding).  But  also  C  is  in  L  (by  hyp.),  hence  (once 
more  by  the  preceding)  A  ©  B  ©  C  is  in  L. 

Scholium.  It  is  obvious  that  these  two  propositions  and  similar  ones  can  be  con- 
verted. For  if  A  ©  B  =  L,  it  is  clear  from  the  definition  of  "inexistent"  that  A  is  in  L, 
and  B  is  in  L.  Likewise,  if  A  ©  B  ©  C  =  L,  it  is  clear  that  A  is  in  L,  and  B  is  in  L,  and 
C  is  in  L.17  Also  that  A  ©  B  is  in  L,  and  A  ©  C  is  in  L,  and  B  ©  C  is  in  L.  And  so  on. 

Prop.  20.  //  A  is  in  M  and  B  is  in  N,  then  A  ©  B  is  in  M  ©  N.  If  the  former  of  one 
pair  be  in  the  latter  and  the  former  of  another  pair  be  in  the  latter,  then  what  is  composed  of  the 
former  in  the  two  cases  is  in  what  is  composed  of  the  latter  in  the  two  cases.  For  A  is  in  M  (by 
hyp.)  and  M  is  in  M  ©  N  (by  def.  of  "inexistent").  Hence  (by  prop.  15)  A  is  in  M  ©  N 
Similarly,  since  B  is  in  N  and  N  is  in  M  ©  N,  then  also  (by  prop.  18)  A  ©  B  is  in  M  ©  N' 


RT  is  in  RY  and  ST  is  in  SX,  hence  RT 
,''"  ^^^^  ©   ST,  or  RY,  is  in  RY  ©  SX,  or  in  RX.18 

^'•"^"X^'^N  If  A  be  "quadrilateral"  and  B  "equi- 

fr          /          \         N>^  angular",  A  ©  5  will  be  "rectangle".     If  M 

f  I  \  ^  be  "parallelogram"  and  N  "regular",  M  © 

;-  -  ^  -  f  -  J£  -  IT          N  will  be  "square".     Now  "quadrilateral" 
V  \     /'  /I  ib  in   [the  intension  of]  "parallelogram",  and 

v\^^-ii--^--_B.-''/  "equiangular"  is  in  "regular",  hence  "rec- 

NXX^  ^'  tangle"    (or  "equiangular  quadrilateral")  is 

m  "regular  parallelogram  or  square". 


Scholium.  This  proposition  cannot  be  converted.  Suppose  that  A  is  in  M  and 
A  ©  B  is  in  M  ©  N,  still  it  does  not  follow  that  B  is  in  N;  for  it  might  happen  that  B  as 
well  as  A  is  in  M,  and  whatever  is  in  B  is  in  M,  and  something  different  in  N.  Much  less, 
therefore,  can  the  following  similar  proposition  be  converted. 

Prop.  21.    IfAisinMandBisinNandCisinP,thenA  ©  B  ©  CisinM  ®  N  @  P. 

17  To  be  consistent,  Leibniz  should  have  written  "A  ©  B  is  in  L"  instead  of  "A  ©  5 
=  L",  and  "A  ©  5  ©  C  is  in  L"  instead  of  "A  .©  £  ©  C  =  L"—  but  note  the  method 
of  the  proof.  to 

18  The  text  has  RY  here  instead  of  BX:  the  correction  is  obvious. 


384  A  Survey  of  Symbolic  Logic 

And  so  on.  Whatever  is  composed  of  terms  which  are  contained,  is  in  what  is  composed  of  the 
containers.  For  since  A  is  in  M  and  B  is  in  N,  (by  the  preceding),  A  ©  B  is  in  M  ©  N. 
But  C  is  in  P,  hence  (again  by  the  preceding)  A®B®C'is'v\M®N@P. 

Prop.  22.  Two  disparate  terms,  A  and  B,  being  given,  to  find  a  third  term,  C,  different 
from  them  and  such  that  with  them  it  composes  subalternates  A  ©  C  and  B  ©  C — that  is, 
such  that  although  A  and  B  are  neither  of  them  contained  in  the  other,  still  A  @  C  and 
B  ©  C  shall  one  of  them  be  contained  in  the  other. 

Solution.  If  we  wish  that  A  ©  C  be  contained  in  B  ©  C,  but  A  be  not  contained  in  B, 
this  can  be  accomplished  in  the  following  manner:  Assume  (by  post.  1)  some  term,  D, 
such  that  it  is  not  contained  in  A,  and  (by  post.  2)  let  A  ©  D  =  C,  and  the  requirements 
are  satisfied. 


-         .. 

/,''  A  ©  C  "-sx  \  For  A  ©  C  =  A  ©  A  ©  D  (by  construc- 

f  \  \  tion)  =  A  ©  D  (by  ax.  2).  Similarly,  B  ©  C 

I  _  S  _  Vr  _  \j£  -  B  ©  A  ©  D  (by  construction).  But  A 

\~  /\  ,/\  f  ©  D  is  in  5  ©  A  ©  D  (by  def.  3).  Hence 

^-*'      X"-j'/'/    N*"~R"'  A  ®  CisinB  ®  C.     Which  was  to  be  done.19 


L> 


SY  and  YX  are  disparate.     If  RS  ©  SY  =  YR,  then  SY  ®  YR  will  be  in  ZF  ©  Ffl. 

Let  A  be  "equilateral",  B  "parallelogram",  D  "equiangular",  and  C  "equiangular 
equilateral"  or  "regular",  where  it  is  obvious  that  although  "equilateral"  and  "parallelo- 
gram" are  disparate,  so  that  neither  is  in  the  other,  yet  "regular  equilateral"  is  in  "regular 
parallelogram "  or  "square ".  But,  you  ask,  will  this  construction  prescribed  in  the  problem 
succeed  in  all  cases?  For  example,  let  A  be  "trilateral",  and  B  "quadrilateral";  is  it 
not  then  impossible  to  find  a  concept  which  shall  contain  A  and  B  both,  and  hence  to 
find  B  ©  C  such  that  it  shall  contain  A  ©  C,  since  A  and  B  are  incompatible?  I  reply 
that  our  general  construction  depends  upon  the  second  postulate,  in  which  is  contained 
the  assumption  that  any  term  and  any  other  term  can  be  put  together  as  components. 
Thus  God,  soul,  body,  point,  and  heat  compose  an  aggregate  of  these  five  things.  And  in 
this  fashion  also  quadrilateral  and  trilateral  can  be  put  together  as  components.  For 
assume  D  to  be  anything  you  please  which  is  not  contained  in  "trilateral",  as  "circle". 
Then  A  ©  D  is  "trilateral  and  circle",20  which  may  be  called  C.  But  C  ©  A  is  nothing 
but  "trilateral  and  circle"  again.  Consequently,  whatever  is  in  C  ©  B  is  also  in  "tri- 
lateral", in  "circle",  and  in  "quadrilateral".  But  if  anyone  wish  to  apply  this  general 
calculus  of  compositions  of  whatever  sort  to  a  special  mode  of  composition;  for  example 
if  one  wish  to  unite  "trilateral"  and  "circle"  and  "quadrilateral"  not  only  to  compose 
an  aggregate  but  so  that  each  of  these  concepts  shall  belong  to  the  same  subject,  then  it  is 
necessary  to  observe  whether  they  are  compatible.  Thus  immovable  straight  lines  at  a 
distance  from  one  another  can  be  added  to  compose  an  aggregate  but  not  to  compose  a 
continuum. 

Prop.  23.  Two  disparate  terms,  A  and  B,  being  given,  to  find  a  third,  C,  different  from 
them  [and  such  that  A  ©  B  =  A  ©  C].21 

Solution.  Assume  (by  post.  2)  C  =  A  ©  B,  and  this  satisfies  the  requirements. 
For  since  A  and  B  are  disparate  (by  hyp.) — that  is  (by  def.  6),  neither  is  in  the  other — 

19  Leibniz  has  carelessly  substituted  L  in  the  proof  where  he  has  D  in  the  proposition 
and  in  the  figure.     We  read  D  throughout. 

20  Leibniz  is  still  sticking  to  intensions  in  this  example,  however  much  the  language 
may  suggest  extension. 

21  The  proof,  as  well  as  the  reference  in  the  scholium  to  prop.  9,  indicate  that  the 
statement  of  the  theorem  in  the  text  is  incomplete.     We  have  chosen  the  most  conservative 
emendation. 


Two  Fragments  from  Leibniz  385 

therefore  (by  prop.  13)  it  is  impossible  that  C  =  A  or  C  =  B.  Hence  these  three  are  differ- 
ent, as  the  problem  requires.  Thus  A®C=A®A®B  (by  construction),  which  (by 
ax.  2)  is  =  A  ®  B.  Therefore  A  ©  C  =  A  ©  B.  Which  was  to  be  done. 

Prop.  24.  To  find  a  set  of  terms,  of  any  desired  number,  which  differ  each  from  each  and 
are  so  related  that  from  them  nothing  can  be  composed  which  is  new,  or  different  from  every 
one  of  them  [i.  e.,  such  that  they  form  a  group  with  respect  to  the  operation  ©]. 

Solution.  Assume  (by  post.  1)  any  terms,  of  any  desired  number,  which  shall  be 
different  from  each  other,  A,  B,  C,  and  D,  and  from  these  let  A  ©  B  =  M,  M  ©  C  =  N, 
and  N  ©  D  —  P.  Then  A,  B,  M,  N,  and  P  are  the  terms  required.  For  (by  construction) 
M  is  made  from  A  and  B,  and  hence  A,  or  B,  is  in  M,  and  M  in  N,  and  N  in  P.  Hence 
(by  prop.  16)  any  term  which  here  precedes  is  in  any  which  follows.  But  if  two  such  are 
united  as  components,  nothing  new  arises;  for  if  a  term  be  united  with  itself,  nothing  new 
arises;  L  ©  L  =  L  (by  ax.  2).22  If  one  term  be  united  with  another  as  components,  a 
term  which  precedes  will  be  united  with  one  which  follows;  hence  a  term  which  is  contained 
with  one  which  contains  it,  as  L  0  N,  but  L  ©  N  =  N  (by  prop.  14) ,23  And  if  three  are 
united,  as  L  ©  N  ©  P,  then  a  couple,  L  ©  N,  will  be  joined  with  one,  P.  But  the  couple, 
L  ©  N,  by  themselves  will  not  compose  anything  new,  but  one  of  themselves,  namely  the 
latter,  N,  as  we  have  shown;  hence  to  unite  a  couple,  L  ©  N,  with  one,  P,  is  the  same  as 
to  unite  one,  N,  with  one,  P,  which  we  have  just  demonstrated  to  compose  nothing  new. 
And  so  on,  for  any  larger  number  of  terms.  Q.E.D. 

Scholium.  It  would  have  been  sufficient  to  add  each  term  to  the  next,  which  contains 
it,  as  M,  N,  P,  etc.,  and  indeed  this  will  be  the  situation,  if  in  our  construction  we  put 
A  =  Nothing  and  let  B  =  M .  But  it  is  clear  that  the  solution  which  has  been  given  is  of 
somewhat  wider  application,  and  of  course  these  problems  can  be  solved  in  more  than  one 
way;  but  to  exhibit  all  their  possible  solutions  would  be  to  demonstrate  that  no  other 
ways  are  possible,  and  for  this  a  large  number  of  propositions  would  need  to  be  proved 
first.  But  to  give  an  example:  five  things,  A,  B,  C,  D,  and  E,  can  be  so  related  that  they 
will  not  compose  anything  new  only  in  some  one  of  the  following  ways:  first,  if  A  is  in  B 
and  B  in  C  and  C  in  D  and  D  in  E;  second,  if  A  ©  B  =  C  and  C  is  in  D  and  D  in  E;  third, 
if  A  ©  B  =  C  and  A  is  in  D  and  B  ©  D  =  E.  The  five  concepts  which  follow  are  related 
in  the  last,  or  third,  way;  A  "equiangular",  B  "equilateral",  C  "regular",  D  "rectangle", 
E  "square",  from  which  nothing  can  be  composed  which  does  not  coincide  with  them,  since 
"equiangular  equilateral"  coincides  with  "regular",  and  "equiangular"  is  in  [the  intension 
of]  "rectangle",  and  "equilateral  rectangle"  coincides  with  "square".  Thus  "regular 
equiangular"  figure  is  the  same  as  that  which  is  at  once  "regular"  and  "regular  equi- 
lateral", and  " equiangular  rectangle "  is  "rectangle",  and  " regular  rectangle "  is  "square". 

Scholium  to  defs.  3,  4,  5,  and  6.  We  say  that  the  concept  of  the  genus  is  in  the  concept 
of  the  spe  :ies;  the  individuals  of  the  species  amongst  (in)  the  individuals  of  the  genus;  a 
part  in  the  whole;  and  indeed  the  ultimate  and  indivisible  in  the  continuous,  as  a  point 
is  in  a  line,  although  a  point  is  not  a  part  of  the  line.  Likewise  the  concept  of  the  attribute 
or  predicate  is  in  the  concept  of  the  subject.  And  in  general  this  conception  is  of  the 
widest  application.  We  also  speak  of  that  which  is  in  something  as  contained  in  that  in 
which  it  is.  We  are  not  here  concerned  with  the  notion  of  "contained"  in  general — with 
the  manner  in  which  those  things  which  are  "in"  are  related  to  one  another  and  to  that 
which  contains  them.  Thus  our  demonstrations  cover  also  those  things  which  compose 
something  in  the  distributive  sense,  as  all  the  species  together  compose  the  genus.  Hence 
all  the  inexistent  things  which  suffice  to  constitute  a  container,  or  in  which  are  all  things 
which  are  in  the  container,  are  said  to  compose  that  container;  as  for  example,  A  ©  B  are 
said  to  compose  L,  if  A,  B,  and  L  denote  the  straight  lines  RS,  YX,  and  RX,  for  RS  ©  YX 
=  RX.  And  such  parts  which  complete  the  whole,  I  am  accustomed  to  call  "cointe- 
grants",  especially  if  they  have  no  common  part;  if  they  have  a  common  part,  they  are 

22  The  number  of  the  axiom  is  omitted  in  the  text. 

23  The  number  of  the  prop,  is  omitted  in  the  text. 

26 


386  A  Survey  of  Symbolic  Logic 

called  "co-members",  as  RS  and  RX.  Whence  it  is  clear  that  the  same  thing  can  be 
composed  in  many  different  ways  if  the  things  of  which  it  is  composed  are  themselves 
composite.  Indeed  if  the  resolution  could  finally  be  carried  to  infinity,  the  variations  of 
composition  would  be  infinite.  Thus  all  synthesis  and  analysis  depends  upon  the  principles 
here  laid  down.  And  if  those  things  which  are  contained  are  homogeneous  with  that  in 
which  they  are  contained,  they  are  called  parts  and  the  container  is  called  the  whole.  If 
two  parts,  however  chosen,  are  such  that  a  third  can  be  found  having  a  part  of  one  and  a 
part  of  the  other  in  common,  then  that  which  is  composed  of  them  is  continuous.  Which 
illustrates  by  what  small  and  simple  additions  one  concept  arises  from  another.  And  I 
call  by  the  name  "subalter  nates"  those  things  one  of  which  is  in  the  other,  as  the  species 
in  the  genus,  the  straight  line  RS  in  the  straight  line  RX;  "disparates"  where  the  opposite 
is  the  case,  as  the  straight  lines  RS  and  YX,  two  species  of  the  same  genus,  perfect  metal 
and  imperfect  metal — and  particularly,  members  of  the  different  divisions  of  the  same 
whole,  which  (members)  have  something  in  common,  as  for  example,  if  you  divide  "metal" 
into  "perfect"  and  "imperfect",  and  again  into  "soluble  in  aquafortis"  and  "insoluble", 
it  is  clear  that  "metal  which  is  insoluble  in  aqua  fortis"  and  "perfect  metal"  are  two  dispa- 
rate things,  and  there  is  metal  which  is  perfect,  or  is  always  capable  of  being  fulminated  in 
a  cupel,24  and  yet  is  soluble  in  aquafortis,  as  silver,  and  on  the  other  hand,  there  is  imperfect 
metal  which  is  insoluble  in  aqua  fortis,  as  tin. 

Scholium  to  axioms  1  and  2.  Since  the  ideal  form  of  the  general  [or  ideal  form  in 
general,  speciosa  generalis]  is  nothing  but  the  representation  of  combinations  by  means  of 
symbols,  and  their  manipulation,  and  the  discoverable  laws  of  combination  are  various,25 
it  results  from  this  that  various  modes  of  computation  arise.  In  this  place,  however,  we 
have  nothing  to  do  with  the  theory  of  the  variations  which  consist  simply  in  changes  of 
order  [i.  e.,  the  theory  of  permutations],  and  AB  [more  consistently,  A  ©  B]  is  for  us  the 
same  as  BA  [or  B  ©  A],  And  also  we  here  take  no  account  of  repetition — that  is  A  A  [more 
consistently,  A  ©  A]  is  for  us  the  same  as  A.  Thus  wherever  these  laws  just  mentioned 
can  be  used,  the  present  calculus  can  be  applied.  It  is  obvious  that  it  can  also  be  used 
in  the  composition  of  absolute  concepts,  where  neither  laws  of  order  nor  of  repetition  obtain; 
thus  to  say  "warm  and  light"  is  the  same  as  to  say  "light  and  warm",  and  to  say  "warm 
fire"  or  "white  mi  k",  after  the  fashion  of  the  poets,  is  pleonasm;  white  milk  is  nothing 
different  from  milk,  and  rational  man — that  is,  rational  animal  which  is  rational — is  nothing 
different  from  rational  animal.  The  same  thing  is  true  when  certain  given  things  are  said 
to  be  contained  in  (inexistere)  certain  things.  For  the  real  addition  of  the  same  is  a  useless 
repetition.  When  two  and  two  are  said  to  make  four,  the  latter  two  must  be  different 
from  the  former.  If  they  were  the  same,  nothing  new  would  arise,  and  it  would  be  as  if 
one  should  in  jest  attempt  to  make  six  eggs  out  of  three  by  first  counting  3  eggs,  then 
taking  away  one  and  counting  the  remaining  2,  and  then  taking  away  one  more  and  counting 
the  remaining  1.  But  in  the  calculus  of  numbers  and  magnitudes,  A  or  B  or  any  other 
symbol  does  not  signify  a  certain  object  but  anything  you  please  with  that  number  of 
congruent  parts,  for  any  two  feet  whatever  are  denoted  by  2;  if  foot  is  the  unit  or  measure, 
then  2  +  2  makes  the  new  thing  4,  and  3  times  3  the  new  thing  9,  for  it  is  presupposed  that 
the  things  added  are  always  different  (although  of  the  same  magnitude) ;  but  the  opposite 
is  the  case  with  certain  things,  as  with  lines.  Suppose  we  describe  by  a  moving  [point] 
the  straight  line,  RY  ©  YX  =  RYX  or  P  ©  B  =  L,  going  from  R  to  X.  If  we  suppose 
this  same  [point]  then  to  return  from  X  to  Y  and  stop  there,  although  it  does  indeed  describe 
YX  or  B  a  second  time,  it  produces  nothing  different  than  if  it  had  described  YX  once. 
Thus  L  ©  B  is  the  same  as  L — that  is,  P  ©  B  ©  B  or  RY  ©  YX  ©  XY  is  the  same  as 
RY  ©  YX.  This  caution  is  of  much  importance  in  making  judgments,  by  means  of 
the  magnitude  and  motion  of  those  things  which  generate26  or  describe,  concerning  the 

24  The  text  here  has".  .  .  fulminabile  persistens  in  capella":  the  correction  is  obvious. 

25 ".  .  .  variaeque  sint  combinandi  leges  excogitabiles,  .  .  ."  "Excogitabiles", 
"discoverable  by  imagination  or  invention",  is  here  significant  of  Leibniz's  theory  of  the 
relation  between  the  "universal  calculus"  and  the  progress  of  science. 

28  Reading  "generant"  for  "generantur" — a  correction  which  is  not  absolutely  neces- 


Two  Fragments  from  Leibniz  387 

magnitude  of  those  things  which  are  generated  or  described.  For  care  must  be  taken  either 
that  one  [step  in  the  process]  shall  not  choose  the  track  of  another  as  its  own — that  is,  one 
part  of  the  describing  operation  follow  in  the  path  of  another — or  else  [if  this  should  happen] 
this  [reduplication]  must  be  subtracted  so  that  the  same  thing  shall  not  be  taken  too  many 
times.  It  is  clear  also  from  this  that  "components",  according  to  the  concept  which  we 
here  use,  can  compose  by  their  magnitudes  a  magnitude  greater  than  the  magnitude  of  the 
thing  which  they  compose.27  Whence  the  composition  of  things  differs  widely  from  the 
composition  of  magnitudes.  For  example,  if  there  are  two  parts,  A  or  RS  and  B  or  RX, 
of  the  whole  line  L  or  RX,  and  each  of  these  is  greater  than  half  of  RX  itself — if,  for  example, 
RX  is  5  feet  and  RS  4  feet  and  YX  3  feet — obviously  the  magnitudes  of  the  parts  compose 
a  magnitude  of  7  feet,  which  is  greater  than  that  of  the  whole;  and  yet  the  lines  RS  and 
YX  themselves  compose  nothing  different  from  RX, — that  is,  RS  ©  YX  =  RX.  Accord- 
ingly I  here  denote  this  real  addition  by  ©,  as  the  addition  of  magnitudes  is  denoted  by  +. 
And  finally,  although  it  is  of  much  importance,  when  it  is  a  question  of  the  actual  generation 
of  things,  what  their  order  is  (for  the  foundations  are  laid  before  the  house  is  built),  still 
in  the  mental  construction  of  things  the  result  is  the  same  whichever  ingredient  we  consider 
first  (although  one  order  may  be  more  convenient  than  another),  hence  the  order  does  not 
here  alter  the  thing  developed.  This  matter  is  to  be  considered  in  its  own  time  and  proper 
place.  For  the  present,  however,  RY  ©  YS  ©  SX  is  the  same  as  YS  ©  RY  ©  SX. 

Scholium  to  prop.  24.     If  RS  and  YX  are  different,  indeed  disparate,  so  that  neither 
is  in  the  other,  then  let  RS  ©  YX  =  RX,  and  RS  ©  RX  will  be  the  same  as  YX  ©  RX 
For  the  straight  line  RX  is  always  composed  by  a  process  of  conception  (in  notionibus). 
If  A  is  "parallelogram"  and  B  "equiangular" 
— which  are  disparate  terms — let  C  be  A  ©  B, 
that    is,    "rectangle".      Then    "rectangular 
parallelogram"  is  the  same  as  "equiangular 
rectangle ",  for  either  of  these  is  nothing  differ- 
ent  from  "rectangle".     In  general,  if  Maevius  ^ 

is  A  and  Titius  B,  the  pair  composed  of  the          j^f y= K. "jP^ 

two  men  C,  then  Maevius  together  with  this  \\x  /  N<k»  JV.^'// 

pair  is  the  same  as  Titius  together  with  this  \\xx  ,''  '/ 

pair,  for  in  either  case  we  have  nothing  more  Vx^~"~~      "'^  ,''/ 

than   the   pair   itself.      Another  solution   of  x^xXv^        Q      ^,''  / 

this  problem,  one  more  elegant  but  less  general,  x^_^  ^^'' 

can  be  given  if  A  and  B  have  something  in  A  9  C 

common,  and  this  common  term  is  given  and 

that  which  is  peculiar  to  each  of  the  terms  A  and  B  is  also  given.     For  let  that  which  is 

exclusively  A  be  M,  and  that  which  is  exclusively  B  be  N,  and  let  M  ©  N  =  D  and  let 

what  is  common  to  A  and  B  be  P.     Then  I  affirm  that  A  ©  D  =  B  ©  D,  for  since  A  =  P 

©  M  and  B  =  P  ©  N,  we  have  A®D  =  P®M@N  and  also  B@D  =  P®M®N. 

sary,  since  a  motion  which  generates  a  line  is  also  itself  generated;  but,  as  the  context  shows, 
"generare"  and  "describere"  are  here  synonymous. 
27  Italics  ours. 


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27 


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Saggio  di  calcolo  geometrico.     Atti  Accad.  Torino,  31  (1896),  pp.  3-26. 

Studii  in  logica  matematica.     Ibid.,  32  (1897),  pp.  3-21. 

Les  definitions  mathematique.    Bill,  du  Cong.  Int.  de  Phil.,  Paris  1900,  3, 

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Arithmetica  generale  e  algebra  elementare.     Torino,  Paravia,  1902. 

La  geometria  basata  sulle  idee  di  punto  e  di  distanza.     Atti  Acad.  Torino,  38 


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*PEANO,   G.,   redacteur.     Formulaire   de   Mathematiques.     Tome   I,    1895;    II,    1897; 
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In  addition  to  Peano,  the  chief  contributor,  the  other  contributors  were  MM. 
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